0132_1203_3102
Counting sequence:
1, 1, 2, 6, 21, 73, 241, 759, 2305, 6806, 19652, 55725, 155688, 429719, 1174344, 3183298, 8571979, 22958381, 61220351, 162668347, 430978487, 1139179386, 3005433282, 7916965441, 20829348046, 54747238203, 143781463846, 377368903574, 989935857665, 2595794110401, 6804425792877, 17831978780015, 46721575315053, 122395024187374, 320592346261984, 839648302566533, 2198902317246644, 5758192520534191, 15078011706559076, 39480652962508122, 103373842785608247, 270661216359422821, 708651587734507387, 1855379308751056851, 4857662260379097571, 12717968112200135858, 33296980948035164126, 87174487659905164089, 228229578256424132250, 597520580296343208371, 1564345110481534168002, 4095541209796070081230, 10722332562102204504493, 28071566814701413991129, 73492593061983405974249, 192406671738410795701543, 503728358901971474171609, 1318780314493745631880326, 3452616475689343469553276, 9039077038909628948826397, 23664630781940607872756032, 61954848165175075964551431, 162199980583031887217992192, 424645229628658129297335986, 1111735985004103606317255971, 2910563288009346937795720413, 7619955022722069777062067831, 19949304104446615680793947179, 52227962012984260134964945935, 136734591526813083053067687466, 357975832047216730861524577402, 937192944164656403564784661745, 2453603080726982688616797990758, 6423616460937935321288369130859, 16817246632652477076123474306014, 44028124107595536474571674075078, 115267127050175679414049369404249, 301773259800793527763449238874161, 790052657943486819593958280169701, 2068384725363346489905574114438047, 5415101541116143222800718382491717, 14176919944528727233659804259151966, 37115658386766252408119770021970072, 97170055406780309489810915410255221, 254394508220430938337994312114689996, 666013470037896436634451726143331743, 1743645903479369046899754343529719836, 4564924243611117680961266392465339594, 11951126833853169202232291277086606095, 31288456271101502843142772859597575061, 81914242006067050171831703212037187827, 214454269800950039381266378734625786435, 561448567505721791429080897182962993723, 1469891432936568661898774001746307193314, 3848225731749642608409978007019643189654, 10074785763213579511930916859439184689513, 26376131559713343665210852335949421611314, 69053608919610561040614931846508873705571, 180784695206565786732974790937370330703226, 473300476715140150036019665108375468829822, 1239116734969278277780561710023676952572677
Generating function in Maple syntax:
-(4*x^8+6*x^7-45*x^6+100*x^5-126*x^4+95*x^3-42*x^2+10*x-1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
-\frac{4 x^{8}+6 x^{7}-45 x^{6}+100 x^{5}-126 x^{4}+95 x^{3}-42 x^{2}+10 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-4*x**8 - 6*x**7 + 45*x**6 - 100*x**5 + 126*x**4 - 95*x**3 + 42*x**2 - 10*x + 1)/((x - 1)**4*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)^2*(x-1)^4*F(x)+4*x^8+6*x^7-45*x^6+100*x^5-126*x^4+95*x^3-42*x^2+10*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+4 x^{8}+6 x^{7}-45 x^{6}+100 x^{5}-126 x^{4}+95 x^{3}-42 x^{2}+10 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/30*(-54*5^(1/2)+180)*(3/2-1/2*5^(1/2))^(-n)+1/30*(54*5^(1/2)+180)*(3/2+1/2*5^(1/2))^(-n)+1/6*n^3-2*n^2-2^n*n+35/6*n-10)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-54 \sqrt{5}+180\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(54 \sqrt{5}+180\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}-2 n^{2}\\-2^{n} n +\frac{35 n}{6}-10 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 73
a(6) = 241
a(7) = 759
a(8) = 2305
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)-1/6*(n-1)*(n^2-14*n+30), n >= 9
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) = 21
a \! \left(5\right) = 73
a \! \left(6\right) = 241
a \! \left(7\right) = 759
a \! \left(8\right) = 2305
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-\frac{\left(n -1\right) \left(n^{2}-14 n +30\right)}{6}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14720/
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[32,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[20,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[19,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[27,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[24,x]*F[4,x]
F[27,x] = F[20,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[18,x]+F[31,x]
F[31,x] = F[29,x]
F[32,x] = F[33,x]+F[47,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[1,x]+F[14,x]
F[37,x] = F[33,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[33,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[41,x]
F[44,x] = F[38,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[44,x]
F[47,x] = F[103,x]+F[21,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[71,x]
F[50,x] = F[51,x]+F[7,x]
F[51,x] = F[21,x]+F[22,x]+F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[17,x]+F[54,x]
F[54,x] = F[55,x]+F[63,x]
F[55,x] = F[21,x]+F[56,x]+F[58,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[24,x]+F[30,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[17,x]+F[60,x]
F[60,x] = F[55,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[60,x]
F[63,x] = 2*F[21,x]+F[62,x]+F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[30,x]+F[66,x]
F[66,x] = F[63,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[61,x]+F[70,x]
F[70,x] = F[68,x]
F[71,x] = F[47,x]+F[72,x]
F[72,x] = 2*F[21,x]+F[73,x]+F[84,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[75,x]+F[78,x]
F[75,x] = F[42,x]+F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[4,x]*F[75,x]
F[78,x] = F[79,x]+F[80,x]
F[79,x] = 2*F[21,x]+F[46,x]+F[73,x]
F[80,x] = F[81,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[45,x]+F[83,x]
F[83,x] = F[81,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[44,x]+F[86,x]
F[86,x] = F[87,x]+F[95,x]
F[87,x] = 2*F[21,x]+F[88,x]+F[90,x]
F[88,x] = F[4,x]*F[89,x]
F[89,x] = F[75,x]+F[82,x]
F[90,x] = F[4,x]*F[91,x]
F[91,x] = F[44,x]+F[92,x]
F[92,x] = F[87,x]+F[93,x]
F[93,x] = F[94,x]
F[94,x] = F[4,x]*F[92,x]
F[95,x] = 3*F[21,x]+F[94,x]+F[96,x]
F[96,x] = F[4,x]*F[97,x]
F[97,x] = F[82,x]+F[98,x]
F[98,x] = F[95,x]+F[99,x]
F[99,x] = F[100,x]
F[100,x] = F[101,x]*F[4,x]
F[101,x] = F[102,x]+F[93,x]
F[102,x] = F[100,x]
F[103,x] = F[104,x]*F[4,x]
F[104,x] = F[105,x]+F[41,x]
F[105,x] = F[106,x]+F[122,x]
F[106,x] = F[107,x]+F[117,x]+F[21,x]
F[107,x] = F[108,x]*F[4,x]
F[108,x] = F[109,x]+F[110,x]
F[109,x] = F[14,x]+F[55,x]
F[110,x] = F[111,x]+F[38,x]
F[111,x] = 2*F[21,x]+F[112,x]+F[88,x]
F[112,x] = F[113,x]*F[4,x]
F[113,x] = F[114,x]+F[44,x]
F[114,x] = F[111,x]+F[115,x]
F[115,x] = F[116,x]
F[116,x] = F[114,x]*F[4,x]
F[117,x] = F[118,x]*F[4,x]
F[118,x] = F[119,x]+F[41,x]
F[119,x] = F[106,x]+F[120,x]
F[120,x] = F[121,x]
F[121,x] = F[119,x]*F[4,x]
F[122,x] = 2*F[21,x]+F[121,x]+F[123,x]
F[123,x] = F[124,x]*F[4,x]
F[124,x] = F[125,x]+F[75,x]
F[125,x] = F[122,x]+F[126,x]
F[126,x] = F[127,x]
F[127,x] = F[128,x]*F[4,x]
F[128,x] = F[120,x]+F[129,x]
F[129,x] = F[127,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_{32}\! \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_{20}\! \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_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{19}\! \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_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{47}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{33}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{38}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{103}\! \left(x \right)+F_{21}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{71}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{7}\! \left(x \right)
F_{51}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{17}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{63}\! \left(x \right)
F_{55}\! \left(x \right) = F_{21}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{24}\! \left(x \right)+F_{30}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{17}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{55}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{62}\! \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_{30}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{63}\! \left(x \right)+F_{67}\! \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_{61}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{68}\! \left(x \right)
F_{71}\! \left(x \right) = F_{47}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{73}\! \left(x \right)+F_{84}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{78}\! \left(x \right)
F_{75}\! \left(x \right) = F_{42}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{80}\! \left(x \right)
F_{79}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{46}\! \left(x \right)+F_{73}\! \left(x \right)
F_{80}\! \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_{45}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{81}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{44}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{95}\! \left(x \right)
F_{87}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{88}\! \left(x \right)+F_{90}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{75}\! \left(x \right)+F_{82}\! \left(x \right)
F_{90}\! \left(x \right) = F_{4}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{44}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{87}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right) F_{92}\! \left(x \right)
F_{95}\! \left(x \right) = 3 F_{21}\! \left(x \right)+F_{94}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{4}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{82}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{95}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{4}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{93}\! \left(x \right)
F_{102}\! \left(x \right) = F_{100}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{41}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{122}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{117}\! \left(x \right)+F_{21}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{4}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{110}\! \left(x \right)
F_{109}\! \left(x \right) = F_{14}\! \left(x \right)+F_{55}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{38}\! \left(x \right)
F_{111}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{112}\! \left(x \right)+F_{88}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{4}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{44}\! \left(x \right)
F_{114}\! \left(x \right) = F_{111}\! \left(x \right)+F_{115}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)
F_{116}\! \left(x \right) = F_{114}\! \left(x \right) F_{4}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right) F_{4}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)+F_{41}\! \left(x \right)
F_{119}\! \left(x \right) = F_{106}\! \left(x \right)+F_{120}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{119}\! \left(x \right) F_{4}\! \left(x \right)
F_{122}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{121}\! \left(x \right)+F_{123}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right) F_{4}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)+F_{75}\! \left(x \right)
F_{125}\! \left(x \right) = F_{122}\! \left(x \right)+F_{126}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right) F_{4}\! \left(x \right)
F_{128}\! \left(x \right) = F_{120}\! \left(x \right)+F_{129}\! \left(x \right)
F_{129}\! \left(x \right) = F_{127}\! \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_32(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_20(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_19(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_27(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_24(x)*F_4(x))
Eq(F_27(x), F_20(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_18(x) + F_31(x))
Eq(F_31(x), F_29(x))
Eq(F_32(x), F_33(x) + F_47(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_1(x) + F_14(x))
Eq(F_37(x), F_33(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_33(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_41(x))
Eq(F_44(x), F_38(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_44(x))
Eq(F_47(x), F_103(x) + F_21(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_71(x))
Eq(F_50(x), F_51(x) + F_7(x))
Eq(F_51(x), F_21(x) + F_22(x) + F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_17(x) + F_54(x))
Eq(F_54(x), F_55(x) + F_63(x))
Eq(F_55(x), F_21(x) + F_56(x) + F_58(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_24(x) + F_30(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_17(x) + F_60(x))
Eq(F_60(x), F_55(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_60(x))
Eq(F_63(x), 2*F_21(x) + F_62(x) + F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_30(x) + F_66(x))
Eq(F_66(x), F_63(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_61(x) + F_70(x))
Eq(F_70(x), F_68(x))
Eq(F_71(x), F_47(x) + F_72(x))
Eq(F_72(x), 2*F_21(x) + F_73(x) + F_84(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_78(x))
Eq(F_75(x), F_42(x) + F_76(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_4(x)*F_75(x))
Eq(F_78(x), F_79(x) + F_80(x))
Eq(F_79(x), 2*F_21(x) + F_46(x) + F_73(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_45(x) + F_83(x))
Eq(F_83(x), F_81(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_44(x) + F_86(x))
Eq(F_86(x), F_87(x) + F_95(x))
Eq(F_87(x), 2*F_21(x) + F_88(x) + F_90(x))
Eq(F_88(x), F_4(x)*F_89(x))
Eq(F_89(x), F_75(x) + F_82(x))
Eq(F_90(x), F_4(x)*F_91(x))
Eq(F_91(x), F_44(x) + F_92(x))
Eq(F_92(x), F_87(x) + F_93(x))
Eq(F_93(x), F_94(x))
Eq(F_94(x), F_4(x)*F_92(x))
Eq(F_95(x), 3*F_21(x) + F_94(x) + F_96(x))
Eq(F_96(x), F_4(x)*F_97(x))
Eq(F_97(x), F_82(x) + F_98(x))
Eq(F_98(x), F_95(x) + F_99(x))
Eq(F_99(x), F_100(x))
Eq(F_100(x), F_101(x)*F_4(x))
Eq(F_101(x), F_102(x) + F_93(x))
Eq(F_102(x), F_100(x))
Eq(F_103(x), F_104(x)*F_4(x))
Eq(F_104(x), F_105(x) + F_41(x))
Eq(F_105(x), F_106(x) + F_122(x))
Eq(F_106(x), F_107(x) + F_117(x) + F_21(x))
Eq(F_107(x), F_108(x)*F_4(x))
Eq(F_108(x), F_109(x) + F_110(x))
Eq(F_109(x), F_14(x) + F_55(x))
Eq(F_110(x), F_111(x) + F_38(x))
Eq(F_111(x), F_112(x) + 2*F_21(x) + F_88(x))
Eq(F_112(x), F_113(x)*F_4(x))
Eq(F_113(x), F_114(x) + F_44(x))
Eq(F_114(x), F_111(x) + F_115(x))
Eq(F_115(x), F_116(x))
Eq(F_116(x), F_114(x)*F_4(x))
Eq(F_117(x), F_118(x)*F_4(x))
Eq(F_118(x), F_119(x) + F_41(x))
Eq(F_119(x), F_106(x) + F_120(x))
Eq(F_120(x), F_121(x))
Eq(F_121(x), F_119(x)*F_4(x))
Eq(F_122(x), F_121(x) + F_123(x) + 2*F_21(x))
Eq(F_123(x), F_124(x)*F_4(x))
Eq(F_124(x), F_125(x) + F_75(x))
Eq(F_125(x), F_122(x) + F_126(x))
Eq(F_126(x), F_127(x))
Eq(F_127(x), F_128(x)*F_4(x))
Eq(F_128(x), F_120(x) + F_129(x))
Eq(F_129(x), F_127(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 0], [1, 0], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 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