0123_0231_1230_2130
Counting sequence:
1, 1, 2, 6, 20, 67, 213, 630, 1740, 4539, 11317, 27242, 63816, 146363, 330157, 734982, 1618860, 3534723, 7662189, 16507978, 35380472, 75486291, 160418357, 339721846, 717205740, 1509925227, 3170865093, 6643743690, 13891495240, 28990983179, 60397924317, 125627732102, 260919193036, 541165799283, 1120986373437, 2319282237162, 4793183386680, 9895604520163, 20409684445349, 42056319600438, 86586540507212, 178120883499931, 366137371828437, 752065953154986, 1543714325129160, 3166593487700187, 6491516650066573, 13299692649225350, 27232703996370540, 55732045388290019, 113997365567359117, 233061280715927562, 476255660593892856, 972777519511446003, 1986087435669760917, 4053239664632769142, 8268608915851501100, 16861477004874352203, 34371472356090782309, 70039981404865049098, 142674036195096343752, 290536219160924400171, 591448731863311389117, 1203650050809547057926, 2448805275784941712780, 4980620899901577588947, 10127262496466542402653, 20586566386259858077610, 41837215559173261443640, 85002596691653612125059, 172661524529921401299717, 350635711353071155181622, 711896747292599013915276, 1445044143758111433222523, 2932589585862049675412597, 5950181768415752966834922, 12070368730214813163650120, 24480747847196240785102843, 49641516467925710483529069, 100643074482917878791293958, 204006232059968673228514092, 413452630308203177746195011, 837785592992938018067892397, 1697331850738939361283806666, 3438185030984005372860516600, 6963412720980264046303535507, 14100910759985034693768601333, 28549992156019082589856612470, 57796325584136191584348210540, 116985333712468435977962368299, 236756032513328977574452410117, 479082795203442166385955742282, 969307050760452755246008692296, 1960897022228042355440206944843, 3966359885870358400776787928477, 8021851454569264181346318618502, 16221966274795623122278117201740, 32800459280905435763727188524083, 66313972024439250565796279221629, 134054050974135259208276356455018, 270960315798784034569920302322232
Generating function in Maple syntax:
(x^7-3*x^6+25*x^5-49*x^4+48*x^3-27*x^2+8*x-1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
\frac{x^{7}-3 x^{6}+25 x^{5}-49 x^{4}+48 x^{3}-27 x^{2}+8 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(x**7 - 3*x**6 + 25*x**5 - 49*x**4 + 48*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**5*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^5*F(x)-x^7+3*x^6-25*x^5+49*x^4-48*x^3+27*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)-x^{7}+3 x^{6}-25 x^{5}+49 x^{4}-48 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,12+9/4*2^n*n-45/4*2^n-1/12*n^4+1/2*n^3+7/12*n^2+6*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 12+\frac{9 \,2^{n} n}{4}-\frac{45 \,2^{n}}{4}-\frac{n^{4}}{12}+\frac{n^{3}}{2}+\frac{7 n^{2}}{12}+6 n & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 67
a(6) = 213
a(7) = 630
a(n+2) = -1/12*n^4+7/6*n^3-29/12*n^2-4*a(n)+4*a(n+1)+7/3*n+1, n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 20
a \! \left(5\right) = 67
a \! \left(6\right) = 213
a \! \left(7\right) = 630
a \! \left(n +2\right) = -\frac{n^{4}}{12}+\frac{7 n^{3}}{6}-\frac{29 n^{2}}{12}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{7 n}{3}+1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15005/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[19,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[11,x]
F[18,x] = F[14,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[202,x]+F[21,x]+F[22,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[45,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[21,x]+F[26,x]+F[38,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[32,x]
F[28,x] = F[11,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[11,x]+F[29,x]
F[32,x] = F[14,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[29,x]
F[37,x] = F[33,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[13,x]+F[40,x]
F[40,x] = F[25,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[18,x]+F[44,x]
F[44,x] = F[41,x]
F[45,x] = F[100,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[51,x]
F[49,x] = F[50,x]+F[7,x]
F[50,x] = F[15,x]
F[51,x] = F[46,x]+F[52,x]
F[52,x] = 2*F[21,x]+F[53,x]+F[72,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[57,x]
F[55,x] = F[50,x]+F[56,x]
F[56,x] = F[42,x]
F[57,x] = F[58,x]+F[63,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[50,x]
F[62,x] = F[58,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[56,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[56,x]
F[71,x] = F[68,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[74,x]+F[99,x]
F[74,x] = F[75,x]
F[75,x] = F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]+F[79,x]
F[78,x] = F[11,x]+F[14,x]
F[79,x] = F[75,x]+F[80,x]
F[80,x] = 2*F[21,x]+F[72,x]+F[81,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[83,x]+F[84,x]
F[83,x] = F[14,x]+F[41,x]
F[84,x] = F[85,x]+F[90,x]
F[85,x] = F[86,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[88,x]+F[89,x]
F[88,x] = F[14,x]
F[89,x] = F[85,x]
F[90,x] = F[91,x]
F[91,x] = F[4,x]*F[92,x]
F[92,x] = F[93,x]+F[94,x]
F[93,x] = F[41,x]
F[94,x] = F[95,x]
F[95,x] = F[4,x]*F[96,x]
F[96,x] = F[97,x]+F[98,x]
F[97,x] = F[41,x]
F[98,x] = F[95,x]
F[99,x] = F[80,x]
F[100,x] = 2*F[21,x]+F[101,x]+F[198,x]
F[101,x] = F[102,x]*F[4,x]
F[102,x] = F[103,x]+F[116,x]
F[103,x] = F[104,x]+F[25,x]
F[104,x] = 2*F[21,x]+F[105,x]+F[109,x]
F[105,x] = F[106,x]*F[4,x]
F[106,x] = F[107,x]+F[108,x]
F[107,x] = F[29,x]
F[108,x] = F[33,x]
F[109,x] = F[110,x]*F[4,x]
F[110,x] = F[111,x]+F[32,x]
F[111,x] = F[112,x]+F[41,x]
F[112,x] = F[113,x]
F[113,x] = F[114,x]*F[4,x]
F[114,x] = F[115,x]+F[37,x]
F[115,x] = F[112,x]
F[116,x] = F[117,x]+F[171,x]
F[117,x] = F[118,x]
F[118,x] = F[119,x]*F[4,x]
F[119,x] = F[120,x]+F[122,x]
F[120,x] = F[121,x]+F[25,x]
F[121,x] = F[34,x]
F[122,x] = F[117,x]+F[123,x]
F[123,x] = 3*F[21,x]+F[124,x]+F[143,x]
F[124,x] = F[125,x]*F[4,x]
F[125,x] = F[126,x]+F[128,x]
F[126,x] = F[121,x]+F[127,x]
F[127,x] = F[113,x]
F[128,x] = F[129,x]+F[134,x]
F[129,x] = F[130,x]
F[130,x] = F[131,x]*F[4,x]
F[131,x] = F[132,x]+F[133,x]
F[132,x] = F[121,x]
F[133,x] = F[129,x]
F[134,x] = F[135,x]
F[135,x] = F[136,x]*F[4,x]
F[136,x] = F[137,x]+F[138,x]
F[137,x] = F[127,x]
F[138,x] = F[139,x]
F[139,x] = F[140,x]*F[4,x]
F[140,x] = F[141,x]+F[142,x]
F[141,x] = F[127,x]
F[142,x] = F[139,x]
F[143,x] = F[144,x]*F[4,x]
F[144,x] = F[145,x]+F[170,x]
F[145,x] = F[146,x]
F[146,x] = F[147,x]
F[147,x] = F[148,x]*F[4,x]
F[148,x] = F[149,x]+F[150,x]
F[149,x] = F[29,x]+F[33,x]
F[150,x] = F[146,x]+F[151,x]
F[151,x] = 3*F[21,x]+F[143,x]+F[152,x]
F[152,x] = F[153,x]*F[4,x]
F[153,x] = F[154,x]+F[155,x]
F[154,x] = F[112,x]+F[33,x]
F[155,x] = F[156,x]+F[161,x]
F[156,x] = F[157,x]
F[157,x] = F[158,x]*F[4,x]
F[158,x] = F[159,x]+F[160,x]
F[159,x] = F[33,x]
F[160,x] = F[156,x]
F[161,x] = F[162,x]
F[162,x] = F[163,x]*F[4,x]
F[163,x] = F[164,x]+F[165,x]
F[164,x] = F[112,x]
F[165,x] = F[166,x]
F[166,x] = F[167,x]*F[4,x]
F[167,x] = F[168,x]+F[169,x]
F[168,x] = F[112,x]
F[169,x] = F[166,x]
F[170,x] = F[151,x]
F[171,x] = 3*F[21,x]+F[172,x]+F[194,x]
F[172,x] = F[173,x]*F[4,x]
F[173,x] = F[174,x]+F[179,x]
F[174,x] = F[104,x]+F[175,x]
F[175,x] = F[176,x]
F[176,x] = F[177,x]*F[4,x]
F[177,x] = F[108,x]+F[178,x]
F[178,x] = F[112,x]
F[179,x] = F[180,x]+F[185,x]
F[180,x] = F[181,x]
F[181,x] = F[182,x]*F[4,x]
F[182,x] = F[183,x]+F[184,x]
F[183,x] = F[104,x]
F[184,x] = F[180,x]
F[185,x] = F[186,x]
F[186,x] = F[187,x]*F[4,x]
F[187,x] = F[188,x]+F[189,x]
F[188,x] = F[175,x]
F[189,x] = F[190,x]
F[190,x] = F[191,x]*F[4,x]
F[191,x] = F[192,x]+F[193,x]
F[192,x] = F[175,x]
F[193,x] = F[190,x]
F[194,x] = F[195,x]*F[4,x]
F[195,x] = F[196,x]+F[197,x]
F[196,x] = F[146,x]
F[197,x] = F[151,x]
F[198,x] = F[199,x]*F[4,x]
F[199,x] = F[200,x]+F[201,x]
F[200,x] = F[146,x]+F[75,x]
F[201,x] = F[151,x]+F[80,x]
F[202,x] = F[203,x]*F[4,x]
F[203,x] = F[204,x]+F[205,x]
F[204,x] = F[2,x]+F[75,x]
F[205,x] = F[20,x]+F[80,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{19}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{202}\! \left(x \right)+F_{21}\! \left(x \right)+F_{22}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{45}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)+F_{38}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{29}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{13}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{25}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{41}\! \left(x \right)
F_{45}\! \left(x \right) = F_{100}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{7}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right)
F_{51}\! \left(x \right) = F_{46}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{53}\! \left(x \right)+F_{72}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{50}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{42}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{63}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{50}\! \left(x \right)
F_{62}\! \left(x \right) = F_{58}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{56}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{56}\! \left(x \right)
F_{71}\! \left(x \right) = F_{68}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{99}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{79}\! \left(x \right)
F_{78}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{79}\! \left(x \right) = F_{75}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{72}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{84}\! \left(x \right)
F_{83}\! \left(x \right) = F_{14}\! \left(x \right)+F_{41}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)+F_{90}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)+F_{89}\! \left(x \right)
F_{88}\! \left(x \right) = F_{14}\! \left(x \right)
F_{89}\! \left(x \right) = F_{85}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{4}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)+F_{94}\! \left(x \right)
F_{93}\! \left(x \right) = F_{41}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{4}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)+F_{98}\! \left(x \right)
F_{97}\! \left(x \right) = F_{41}\! \left(x \right)
F_{98}\! \left(x \right) = F_{95}\! \left(x \right)
F_{99}\! \left(x \right) = F_{80}\! \left(x \right)
F_{100}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{101}\! \left(x \right)+F_{198}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{4}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{116}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{25}\! \left(x \right)
F_{104}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{105}\! \left(x \right)+F_{109}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{4}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{108}\! \left(x \right)
F_{107}\! \left(x \right) = F_{29}\! \left(x \right)
F_{108}\! \left(x \right) = F_{33}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{4}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{32}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)+F_{41}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right) F_{4}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{37}\! \left(x \right)
F_{115}\! \left(x \right) = F_{112}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{171}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right) F_{4}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)+F_{122}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)+F_{25}\! \left(x \right)
F_{121}\! \left(x \right) = F_{34}\! \left(x \right)
F_{122}\! \left(x \right) = F_{117}\! \left(x \right)+F_{123}\! \left(x \right)
F_{123}\! \left(x \right) = 3 F_{21}\! \left(x \right)+F_{124}\! \left(x \right)+F_{143}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right) F_{4}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{128}\! \left(x \right)
F_{126}\! \left(x \right) = F_{121}\! \left(x \right)+F_{127}\! \left(x \right)
F_{127}\! \left(x \right) = F_{113}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)+F_{134}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right) F_{4}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)+F_{133}\! \left(x \right)
F_{132}\! \left(x \right) = F_{121}\! \left(x \right)
F_{133}\! \left(x \right) = F_{129}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right) F_{4}\! \left(x \right)
F_{136}\! \left(x \right) = F_{137}\! \left(x \right)+F_{138}\! \left(x \right)
F_{137}\! \left(x \right) = F_{127}\! \left(x \right)
F_{138}\! \left(x \right) = F_{139}\! \left(x \right)
F_{139}\! \left(x \right) = F_{140}\! \left(x \right) F_{4}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right)+F_{142}\! \left(x \right)
F_{141}\! \left(x \right) = F_{127}\! \left(x \right)
F_{142}\! \left(x \right) = F_{139}\! \left(x \right)
F_{143}\! \left(x \right) = F_{144}\! \left(x \right) F_{4}\! \left(x \right)
F_{144}\! \left(x \right) = F_{145}\! \left(x \right)+F_{170}\! \left(x \right)
F_{145}\! \left(x \right) = F_{146}\! \left(x \right)
F_{146}\! \left(x \right) = F_{147}\! \left(x \right)
F_{147}\! \left(x \right) = F_{148}\! \left(x \right) F_{4}\! \left(x \right)
F_{148}\! \left(x \right) = F_{149}\! \left(x \right)+F_{150}\! \left(x \right)
F_{149}\! \left(x \right) = F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{150}\! \left(x \right) = F_{146}\! \left(x \right)+F_{151}\! \left(x \right)
F_{151}\! \left(x \right) = 3 F_{21}\! \left(x \right)+F_{143}\! \left(x \right)+F_{152}\! \left(x \right)
F_{152}\! \left(x \right) = F_{153}\! \left(x \right) F_{4}\! \left(x \right)
F_{153}\! \left(x \right) = F_{154}\! \left(x \right)+F_{155}\! \left(x \right)
F_{154}\! \left(x \right) = F_{112}\! \left(x \right)+F_{33}\! \left(x \right)
F_{155}\! \left(x \right) = F_{156}\! \left(x \right)+F_{161}\! \left(x \right)
F_{156}\! \left(x \right) = F_{157}\! \left(x \right)
F_{157}\! \left(x \right) = F_{158}\! \left(x \right) F_{4}\! \left(x \right)
F_{158}\! \left(x \right) = F_{159}\! \left(x \right)+F_{160}\! \left(x \right)
F_{159}\! \left(x \right) = F_{33}\! \left(x \right)
F_{160}\! \left(x \right) = F_{156}\! \left(x \right)
F_{161}\! \left(x \right) = F_{162}\! \left(x \right)
F_{162}\! \left(x \right) = F_{163}\! \left(x \right) F_{4}\! \left(x \right)
F_{163}\! \left(x \right) = F_{164}\! \left(x \right)+F_{165}\! \left(x \right)
F_{164}\! \left(x \right) = F_{112}\! \left(x \right)
F_{165}\! \left(x \right) = F_{166}\! \left(x \right)
F_{166}\! \left(x \right) = F_{167}\! \left(x \right) F_{4}\! \left(x \right)
F_{167}\! \left(x \right) = F_{168}\! \left(x \right)+F_{169}\! \left(x \right)
F_{168}\! \left(x \right) = F_{112}\! \left(x \right)
F_{169}\! \left(x \right) = F_{166}\! \left(x \right)
F_{170}\! \left(x \right) = F_{151}\! \left(x \right)
F_{171}\! \left(x \right) = 3 F_{21}\! \left(x \right)+F_{172}\! \left(x \right)+F_{194}\! \left(x \right)
F_{172}\! \left(x \right) = F_{173}\! \left(x \right) F_{4}\! \left(x \right)
F_{173}\! \left(x \right) = F_{174}\! \left(x \right)+F_{179}\! \left(x \right)
F_{174}\! \left(x \right) = F_{104}\! \left(x \right)+F_{175}\! \left(x \right)
F_{175}\! \left(x \right) = F_{176}\! \left(x \right)
F_{176}\! \left(x \right) = F_{177}\! \left(x \right) F_{4}\! \left(x \right)
F_{177}\! \left(x \right) = F_{108}\! \left(x \right)+F_{178}\! \left(x \right)
F_{178}\! \left(x \right) = F_{112}\! \left(x \right)
F_{179}\! \left(x \right) = F_{180}\! \left(x \right)+F_{185}\! \left(x \right)
F_{180}\! \left(x \right) = F_{181}\! \left(x \right)
F_{181}\! \left(x \right) = F_{182}\! \left(x \right) F_{4}\! \left(x \right)
F_{182}\! \left(x \right) = F_{183}\! \left(x \right)+F_{184}\! \left(x \right)
F_{183}\! \left(x \right) = F_{104}\! \left(x \right)
F_{184}\! \left(x \right) = F_{180}\! \left(x \right)
F_{185}\! \left(x \right) = F_{186}\! \left(x \right)
F_{186}\! \left(x \right) = F_{187}\! \left(x \right) F_{4}\! \left(x \right)
F_{187}\! \left(x \right) = F_{188}\! \left(x \right)+F_{189}\! \left(x \right)
F_{188}\! \left(x \right) = F_{175}\! \left(x \right)
F_{189}\! \left(x \right) = F_{190}\! \left(x \right)
F_{190}\! \left(x \right) = F_{191}\! \left(x \right) F_{4}\! \left(x \right)
F_{191}\! \left(x \right) = F_{192}\! \left(x \right)+F_{193}\! \left(x \right)
F_{192}\! \left(x \right) = F_{175}\! \left(x \right)
F_{193}\! \left(x \right) = F_{190}\! \left(x \right)
F_{194}\! \left(x \right) = F_{195}\! \left(x \right) F_{4}\! \left(x \right)
F_{195}\! \left(x \right) = F_{196}\! \left(x \right)+F_{197}\! \left(x \right)
F_{196}\! \left(x \right) = F_{146}\! \left(x \right)
F_{197}\! \left(x \right) = F_{151}\! \left(x \right)
F_{198}\! \left(x \right) = F_{199}\! \left(x \right) F_{4}\! \left(x \right)
F_{199}\! \left(x \right) = F_{200}\! \left(x \right)+F_{201}\! \left(x \right)
F_{200}\! \left(x \right) = F_{146}\! \left(x \right)+F_{75}\! \left(x \right)
F_{201}\! \left(x \right) = F_{151}\! \left(x \right)+F_{80}\! \left(x \right)
F_{202}\! \left(x \right) = F_{203}\! \left(x \right) F_{4}\! \left(x \right)
F_{203}\! \left(x \right) = F_{204}\! \left(x \right)+F_{205}\! \left(x \right)
F_{204}\! \left(x \right) = F_{2}\! \left(x \right)+F_{75}\! \left(x \right)
F_{205}\! \left(x \right) = F_{20}\! \left(x \right)+F_{80}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_19(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_202(x) + F_21(x) + F_22(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_45(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_21(x) + F_26(x) + F_38(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_32(x))
Eq(F_28(x), F_11(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_11(x) + F_29(x))
Eq(F_32(x), F_14(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_29(x))
Eq(F_37(x), F_33(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_13(x) + F_40(x))
Eq(F_40(x), F_25(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_18(x) + F_44(x))
Eq(F_44(x), F_41(x))
Eq(F_45(x), F_100(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_51(x))
Eq(F_49(x), F_50(x) + F_7(x))
Eq(F_50(x), F_15(x))
Eq(F_51(x), F_46(x) + F_52(x))
Eq(F_52(x), 2*F_21(x) + F_53(x) + F_72(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_57(x))
Eq(F_55(x), F_50(x) + F_56(x))
Eq(F_56(x), F_42(x))
Eq(F_57(x), F_58(x) + F_63(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_50(x))
Eq(F_62(x), F_58(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_67(x))
Eq(F_66(x), F_56(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_56(x))
Eq(F_71(x), F_68(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_99(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_79(x))
Eq(F_78(x), F_11(x) + F_14(x))
Eq(F_79(x), F_75(x) + F_80(x))
Eq(F_80(x), 2*F_21(x) + F_72(x) + F_81(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_84(x))
Eq(F_83(x), F_14(x) + F_41(x))
Eq(F_84(x), F_85(x) + F_90(x))
Eq(F_85(x), F_86(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_88(x) + F_89(x))
Eq(F_88(x), F_14(x))
Eq(F_89(x), F_85(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), F_4(x)*F_92(x))
Eq(F_92(x), F_93(x) + F_94(x))
Eq(F_93(x), F_41(x))
Eq(F_94(x), F_95(x))
Eq(F_95(x), F_4(x)*F_96(x))
Eq(F_96(x), F_97(x) + F_98(x))
Eq(F_97(x), F_41(x))
Eq(F_98(x), F_95(x))
Eq(F_99(x), F_80(x))
Eq(F_100(x), F_101(x) + F_198(x) + 2*F_21(x))
Eq(F_101(x), F_102(x)*F_4(x))
Eq(F_102(x), F_103(x) + F_116(x))
Eq(F_103(x), F_104(x) + F_25(x))
Eq(F_104(x), F_105(x) + F_109(x) + 2*F_21(x))
Eq(F_105(x), F_106(x)*F_4(x))
Eq(F_106(x), F_107(x) + F_108(x))
Eq(F_107(x), F_29(x))
Eq(F_108(x), F_33(x))
Eq(F_109(x), F_110(x)*F_4(x))
Eq(F_110(x), F_111(x) + F_32(x))
Eq(F_111(x), F_112(x) + F_41(x))
Eq(F_112(x), F_113(x))
Eq(F_113(x), F_114(x)*F_4(x))
Eq(F_114(x), F_115(x) + F_37(x))
Eq(F_115(x), F_112(x))
Eq(F_116(x), F_117(x) + F_171(x))
Eq(F_117(x), F_118(x))
Eq(F_118(x), F_119(x)*F_4(x))
Eq(F_119(x), F_120(x) + F_122(x))
Eq(F_120(x), F_121(x) + F_25(x))
Eq(F_121(x), F_34(x))
Eq(F_122(x), F_117(x) + F_123(x))
Eq(F_123(x), F_124(x) + F_143(x) + 3*F_21(x))
Eq(F_124(x), F_125(x)*F_4(x))
Eq(F_125(x), F_126(x) + F_128(x))
Eq(F_126(x), F_121(x) + F_127(x))
Eq(F_127(x), F_113(x))
Eq(F_128(x), F_129(x) + F_134(x))
Eq(F_129(x), F_130(x))
Eq(F_130(x), F_131(x)*F_4(x))
Eq(F_131(x), F_132(x) + F_133(x))
Eq(F_132(x), F_121(x))
Eq(F_133(x), F_129(x))
Eq(F_134(x), F_135(x))
Eq(F_135(x), F_136(x)*F_4(x))
Eq(F_136(x), F_137(x) + F_138(x))
Eq(F_137(x), F_127(x))
Eq(F_138(x), F_139(x))
Eq(F_139(x), F_140(x)*F_4(x))
Eq(F_140(x), F_141(x) + F_142(x))
Eq(F_141(x), F_127(x))
Eq(F_142(x), F_139(x))
Eq(F_143(x), F_144(x)*F_4(x))
Eq(F_144(x), F_145(x) + F_170(x))
Eq(F_145(x), F_146(x))
Eq(F_146(x), F_147(x))
Eq(F_147(x), F_148(x)*F_4(x))
Eq(F_148(x), F_149(x) + F_150(x))
Eq(F_149(x), F_29(x) + F_33(x))
Eq(F_150(x), F_146(x) + F_151(x))
Eq(F_151(x), F_143(x) + F_152(x) + 3*F_21(x))
Eq(F_152(x), F_153(x)*F_4(x))
Eq(F_153(x), F_154(x) + F_155(x))
Eq(F_154(x), F_112(x) + F_33(x))
Eq(F_155(x), F_156(x) + F_161(x))
Eq(F_156(x), F_157(x))
Eq(F_157(x), F_158(x)*F_4(x))
Eq(F_158(x), F_159(x) + F_160(x))
Eq(F_159(x), F_33(x))
Eq(F_160(x), F_156(x))
Eq(F_161(x), F_162(x))
Eq(F_162(x), F_163(x)*F_4(x))
Eq(F_163(x), F_164(x) + F_165(x))
Eq(F_164(x), F_112(x))
Eq(F_165(x), F_166(x))
Eq(F_166(x), F_167(x)*F_4(x))
Eq(F_167(x), F_168(x) + F_169(x))
Eq(F_168(x), F_112(x))
Eq(F_169(x), F_166(x))
Eq(F_170(x), F_151(x))
Eq(F_171(x), F_172(x) + F_194(x) + 3*F_21(x))
Eq(F_172(x), F_173(x)*F_4(x))
Eq(F_173(x), F_174(x) + F_179(x))
Eq(F_174(x), F_104(x) + F_175(x))
Eq(F_175(x), F_176(x))
Eq(F_176(x), F_177(x)*F_4(x))
Eq(F_177(x), F_108(x) + F_178(x))
Eq(F_178(x), F_112(x))
Eq(F_179(x), F_180(x) + F_185(x))
Eq(F_180(x), F_181(x))
Eq(F_181(x), F_182(x)*F_4(x))
Eq(F_182(x), F_183(x) + F_184(x))
Eq(F_183(x), F_104(x))
Eq(F_184(x), F_180(x))
Eq(F_185(x), F_186(x))
Eq(F_186(x), F_187(x)*F_4(x))
Eq(F_187(x), F_188(x) + F_189(x))
Eq(F_188(x), F_175(x))
Eq(F_189(x), F_190(x))
Eq(F_190(x), F_191(x)*F_4(x))
Eq(F_191(x), F_192(x) + F_193(x))
Eq(F_192(x), F_175(x))
Eq(F_193(x), F_190(x))
Eq(F_194(x), F_195(x)*F_4(x))
Eq(F_195(x), F_196(x) + F_197(x))
Eq(F_196(x), F_146(x))
Eq(F_197(x), F_151(x))
Eq(F_198(x), F_199(x)*F_4(x))
Eq(F_199(x), F_200(x) + F_201(x))
Eq(F_200(x), F_146(x) + F_75(x))
Eq(F_201(x), F_151(x) + F_80(x))
Eq(F_202(x), F_203(x)*F_4(x))
Eq(F_203(x), F_204(x) + F_205(x))
Eq(F_204(x), F_2(x) + F_75(x))
Eq(F_205(x), F_20(x) + F_80(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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