0132_2130_2301
Counting sequence:
1, 1, 2, 6, 21, 71, 220, 630, 1697, 4365, 10842, 26216, 62071, 144519, 331928, 753834, 1695933, 3784913, 8388838, 18481404, 40501523, 88342827, 191889732, 415236446, 895484281, 1925185941, 4127195570, 8824816080, 18824036847, 40063992335, 85094040112, 180388627026, 381715219061, 806380110489, 1700807049918, 3582002725604, 7533372637963, 15822659519283, 33191507264348, 69544110457734, 145547851727793, 304289842987997, 635517720855562, 1326011023098936, 2764172232229991, 5757042883036311, 11980278696248520, 24910535438894330, 51756211342673197, 107453072359294305, 222928181554840982, 462181911758898636, 957577870769653251, 1982709735949862459, 4102779260534523508, 8484781697966016174, 17537016948980713193, 36226955402568271653, 74795782611369199458, 154347366429241640864, 318350450459565623263, 656300566497447643167, 1352376924903831504992, 2785458355130142296226, 5734631563914456860901, 11801304521155685648681, 24275915201001769929070, 49916889463458046675380, 102600790537972526090747, 210809391274352755870275, 432981976898110595533452, 888985490400210711481046, 1824604349818759169444641, 3743656029294911243160429, 7678567901146043117472698, 15744369853887397142465544, 32272652543931155390401623, 66132020226106511572602023, 135475249660564381890514168, 277448475601557395595070794, 568044019491423883465067933, 1162684407014369608773668337, 2379166013001590215821758022, 4867135349768497057367068316, 9954295198706856624530657011, 20353475099031896785353183051, 41606391007857077676687757220, 85031006448414557632133599230, 173737147388457587955373969497, 354901935012627457560142680245, 724813893001590150953437237522, 1479957316965672118641903014256, 3021192665875970560891312673231, 6166179335680479149272537764399, 12582422559296605114074698617488, 25669924654621645380308239913714, 52359911901614444107133358183253, 106779756028599760991698859070393, 217718990589198399707058775528478, 443836166404908819199033209787524, 904627159587870206643084824941803
Generating function in Maple syntax:
(2*x^7-16*x^5+36*x^4-42*x^3+26*x^2-8*x+1)/(2*x-1)^3/(x-1)^3
Generating function in latex syntax:
\frac{2 x^{7}-16 x^{5}+36 x^{4}-42 x^{3}+26 x^{2}-8 x +1}{\left(2 x -1\right)^{3} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(2*x**7 - 16*x**5 + 36*x**4 - 42*x**3 + 26*x**2 - 8*x + 1)/((x - 1)**3*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^3*(x-1)^3*F(x)+2*x^7-16*x^5+36*x^4-42*x^3+26*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{3} \left(x -1\right)^{3} F \! \left(x \right)+2 x^{7}-16 x^{5}+36 x^{4}-42 x^{3}+26 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/16*(n^2+15*n-82)*2^n+1/2*n^2+7/2*n+5)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(n^{2}+15 n -82\right) 2^{n}}{16}+\frac{n^{2}}{2}+\frac{7 n}{2}+5 & \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) = 71
a(6) = 220
a(7) = 630
a(n+3) = -1/2*n^2+8*a(n)-12*a(n+1)+6*a(n+2)-1/2*n+4, 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) = 21
a \! \left(5\right) = 71
a \! \left(6\right) = 220
a \! \left(7\right) = 630
a \! \left(n +3\right) = -\frac{n^{2}}{2}+8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-\frac{n}{2}+4, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14741/
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[12,x]*F[4,x]
F[4,x] = F[42,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[25,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[1,x]+F[21,x]
F[24,x] = F[19,x]
F[25,x] = F[26,x]+F[27,x]+F[39,x]
F[26,x] = 0
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[10,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[29,x]
F[32,x] = F[25,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[41,x]
F[36,x] = F[26,x]+F[37,x]+F[39,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[10,x]+F[36,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[21,x]+F[36,x]
F[41,x] = F[34,x]
F[42,x] = F[43,x]+F[60,x]
F[43,x] = F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[1,x]+F[14,x]
F[47,x] = F[43,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[20,x]+F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[49,x]
F[53,x] = F[54,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[59,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[21,x]+F[56,x]
F[59,x] = F[54,x]
F[60,x] = F[128,x]+F[26,x]+F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]+F[90,x]
F[63,x] = F[6,x]+F[64,x]
F[64,x] = F[26,x]+F[27,x]+F[65,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[40,x]+F[67,x]
F[67,x] = F[68,x]+F[75,x]
F[68,x] = F[26,x]+F[69,x]+F[71,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[10,x]+F[24,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[20,x]+F[73,x]
F[73,x] = F[68,x]+F[74,x]
F[74,x] = F[54,x]
F[75,x] = F[26,x]+F[76,x]+F[78,x]+F[87,x]
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[30,x]+F[41,x]
F[78,x] = F[12,x]*F[79,x]
F[79,x] = F[35,x]+F[80,x]
F[80,x] = F[75,x]+F[81,x]
F[81,x] = F[82,x]
F[82,x] = F[12,x]*F[83,x]
F[83,x] = F[84,x]+F[89,x]
F[84,x] = 2*F[26,x]+F[85,x]+F[87,x]
F[85,x] = F[12,x]*F[86,x]
F[86,x] = F[36,x]+F[84,x]
F[87,x] = F[12,x]*F[88,x]
F[88,x] = F[56,x]+F[84,x]
F[89,x] = F[82,x]
F[90,x] = F[60,x]+F[91,x]
F[91,x] = 2*F[26,x]+F[92,x]+F[97,x]
F[92,x] = F[12,x]*F[93,x]
F[93,x] = F[35,x]+F[94,x]
F[94,x] = F[95,x]+F[96,x]
F[95,x] = 2*F[26,x]+F[87,x]+F[92,x]
F[96,x] = F[82,x]
F[97,x] = F[12,x]*F[98,x]
F[98,x] = F[88,x]+F[99,x]
F[99,x] = F[100,x]+F[113,x]
F[100,x] = 2*F[26,x]+F[101,x]+F[103,x]
F[101,x] = F[102,x]*F[12,x]
F[102,x] = F[24,x]+F[59,x]
F[103,x] = F[104,x]*F[12,x]
F[104,x] = F[105,x]+F[55,x]
F[105,x] = F[100,x]+F[106,x]
F[106,x] = F[107,x]
F[107,x] = F[108,x]*F[12,x]
F[108,x] = F[109,x]+F[112,x]
F[109,x] = F[110,x]
F[110,x] = F[111,x]*F[12,x]
F[111,x] = F[109,x]+F[56,x]
F[112,x] = F[107,x]
F[113,x] = 2*F[26,x]+F[114,x]+F[116,x]+F[125,x]
F[114,x] = F[115,x]*F[12,x]
F[115,x] = F[41,x]+F[89,x]
F[116,x] = F[117,x]*F[12,x]
F[117,x] = F[118,x]+F[83,x]
F[118,x] = F[113,x]+F[119,x]
F[119,x] = F[120,x]
F[120,x] = F[12,x]*F[121,x]
F[121,x] = F[122,x]+F[127,x]
F[122,x] = 3*F[26,x]+F[123,x]+F[125,x]
F[123,x] = F[12,x]*F[124,x]
F[124,x] = F[122,x]+F[84,x]
F[125,x] = F[12,x]*F[126,x]
F[126,x] = F[109,x]+F[122,x]
F[127,x] = F[120,x]
F[128,x] = F[12,x]*F[129,x]
F[129,x] = F[130,x]+F[40,x]
F[130,x] = F[131,x]+F[138,x]
F[131,x] = F[132,x]+F[134,x]+F[26,x]
F[132,x] = F[12,x]*F[133,x]
F[133,x] = F[14,x]+F[52,x]
F[134,x] = F[12,x]*F[135,x]
F[135,x] = F[136,x]+F[20,x]
F[136,x] = F[131,x]+F[137,x]
F[137,x] = F[54,x]
F[138,x] = F[139,x]+F[141,x]+F[26,x]+F[87,x]
F[139,x] = F[12,x]*F[140,x]
F[140,x] = F[25,x]+F[95,x]
F[141,x] = F[12,x]*F[142,x]
F[142,x] = F[143,x]+F[35,x]
F[143,x] = F[138,x]+F[144,x]
F[144,x] = F[82,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{42}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \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_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{39}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{25}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{36}\! \left(x \right) = F_{26}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right)+F_{36}\! \left(x \right)
F_{41}\! \left(x \right) = F_{34}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{60}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{20}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{49}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{54}\! \left(x \right)
F_{60}\! \left(x \right) = F_{128}\! \left(x \right)+F_{26}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{90}\! \left(x \right)
F_{63}\! \left(x \right) = F_{6}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{40}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{75}\! \left(x \right)
F_{68}\! \left(x \right) = F_{26}\! \left(x \right)+F_{69}\! \left(x \right)+F_{71}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{20}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{68}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{54}\! \left(x \right)
F_{75}\! \left(x \right) = F_{26}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)+F_{87}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{30}\! \left(x \right)+F_{41}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{35}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{75}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{89}\! \left(x \right)
F_{84}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{85}\! \left(x \right)+F_{87}\! \left(x \right)
F_{85}\! \left(x \right) = F_{12}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{36}\! \left(x \right)+F_{84}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{56}\! \left(x \right)+F_{84}\! \left(x \right)
F_{89}\! \left(x \right) = F_{82}\! \left(x \right)
F_{90}\! \left(x \right) = F_{60}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{92}\! \left(x \right)+F_{97}\! \left(x \right)
F_{92}\! \left(x \right) = F_{12}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{35}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)+F_{96}\! \left(x \right)
F_{95}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{87}\! \left(x \right)+F_{92}\! \left(x \right)
F_{96}\! \left(x \right) = F_{82}\! \left(x \right)
F_{97}\! \left(x \right) = F_{12}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{88}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{113}\! \left(x \right)
F_{100}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{101}\! \left(x \right)+F_{103}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{12}\! \left(x \right)
F_{102}\! \left(x \right) = F_{24}\! \left(x \right)+F_{59}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{12}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{55}\! \left(x \right)
F_{105}\! \left(x \right) = F_{100}\! \left(x \right)+F_{106}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{12}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{112}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right) F_{12}\! \left(x \right)
F_{111}\! \left(x \right) = F_{109}\! \left(x \right)+F_{56}\! \left(x \right)
F_{112}\! \left(x \right) = F_{107}\! \left(x \right)
F_{113}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{114}\! \left(x \right)+F_{116}\! \left(x \right)+F_{125}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right) F_{12}\! \left(x \right)
F_{115}\! \left(x \right) = F_{41}\! \left(x \right)+F_{89}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right) F_{12}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)+F_{83}\! \left(x \right)
F_{118}\! \left(x \right) = F_{113}\! \left(x \right)+F_{119}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)
F_{120}\! \left(x \right) = F_{12}\! \left(x \right) F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)+F_{127}\! \left(x \right)
F_{122}\! \left(x \right) = 3 F_{26}\! \left(x \right)+F_{123}\! \left(x \right)+F_{125}\! \left(x \right)
F_{123}\! \left(x \right) = F_{12}\! \left(x \right) F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = F_{122}\! \left(x \right)+F_{84}\! \left(x \right)
F_{125}\! \left(x \right) = F_{12}\! \left(x \right) F_{126}\! \left(x \right)
F_{126}\! \left(x \right) = F_{109}\! \left(x \right)+F_{122}\! \left(x \right)
F_{127}\! \left(x \right) = F_{120}\! \left(x \right)
F_{128}\! \left(x \right) = F_{12}\! \left(x \right) F_{129}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)+F_{40}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right)+F_{138}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)+F_{134}\! \left(x \right)+F_{26}\! \left(x \right)
F_{132}\! \left(x \right) = F_{12}\! \left(x \right) F_{133}\! \left(x \right)
F_{133}\! \left(x \right) = F_{14}\! \left(x \right)+F_{52}\! \left(x \right)
F_{134}\! \left(x \right) = F_{12}\! \left(x \right) F_{135}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)+F_{20}\! \left(x \right)
F_{136}\! \left(x \right) = F_{131}\! \left(x \right)+F_{137}\! \left(x \right)
F_{137}\! \left(x \right) = F_{54}\! \left(x \right)
F_{138}\! \left(x \right) = F_{139}\! \left(x \right)+F_{141}\! \left(x \right)+F_{26}\! \left(x \right)+F_{87}\! \left(x \right)
F_{139}\! \left(x \right) = F_{12}\! \left(x \right) F_{140}\! \left(x \right)
F_{140}\! \left(x \right) = F_{25}\! \left(x \right)+F_{95}\! \left(x \right)
F_{141}\! \left(x \right) = F_{12}\! \left(x \right) F_{142}\! \left(x \right)
F_{142}\! \left(x \right) = F_{143}\! \left(x \right)+F_{35}\! \left(x \right)
F_{143}\! \left(x \right) = F_{138}\! \left(x \right)+F_{144}\! \left(x \right)
F_{144}\! \left(x \right) = F_{82}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_42(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_25(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_1(x) + F_21(x))
Eq(F_24(x), F_19(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_39(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_10(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_29(x))
Eq(F_32(x), F_25(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_41(x))
Eq(F_36(x), F_26(x) + F_37(x) + F_39(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_10(x) + F_36(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_21(x) + F_36(x))
Eq(F_41(x), F_34(x))
Eq(F_42(x), F_43(x) + F_60(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_1(x) + F_14(x))
Eq(F_47(x), F_43(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_20(x) + F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_49(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_59(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_21(x) + F_56(x))
Eq(F_59(x), F_54(x))
Eq(F_60(x), F_128(x) + F_26(x) + F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_90(x))
Eq(F_63(x), F_6(x) + F_64(x))
Eq(F_64(x), F_26(x) + F_27(x) + F_65(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_40(x) + F_67(x))
Eq(F_67(x), F_68(x) + F_75(x))
Eq(F_68(x), F_26(x) + F_69(x) + F_71(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_10(x) + F_24(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_20(x) + F_73(x))
Eq(F_73(x), F_68(x) + F_74(x))
Eq(F_74(x), F_54(x))
Eq(F_75(x), F_26(x) + F_76(x) + F_78(x) + F_87(x))
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_30(x) + F_41(x))
Eq(F_78(x), F_12(x)*F_79(x))
Eq(F_79(x), F_35(x) + F_80(x))
Eq(F_80(x), F_75(x) + F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_12(x)*F_83(x))
Eq(F_83(x), F_84(x) + F_89(x))
Eq(F_84(x), 2*F_26(x) + F_85(x) + F_87(x))
Eq(F_85(x), F_12(x)*F_86(x))
Eq(F_86(x), F_36(x) + F_84(x))
Eq(F_87(x), F_12(x)*F_88(x))
Eq(F_88(x), F_56(x) + F_84(x))
Eq(F_89(x), F_82(x))
Eq(F_90(x), F_60(x) + F_91(x))
Eq(F_91(x), 2*F_26(x) + F_92(x) + F_97(x))
Eq(F_92(x), F_12(x)*F_93(x))
Eq(F_93(x), F_35(x) + F_94(x))
Eq(F_94(x), F_95(x) + F_96(x))
Eq(F_95(x), 2*F_26(x) + F_87(x) + F_92(x))
Eq(F_96(x), F_82(x))
Eq(F_97(x), F_12(x)*F_98(x))
Eq(F_98(x), F_88(x) + F_99(x))
Eq(F_99(x), F_100(x) + F_113(x))
Eq(F_100(x), F_101(x) + F_103(x) + 2*F_26(x))
Eq(F_101(x), F_102(x)*F_12(x))
Eq(F_102(x), F_24(x) + F_59(x))
Eq(F_103(x), F_104(x)*F_12(x))
Eq(F_104(x), F_105(x) + F_55(x))
Eq(F_105(x), F_100(x) + F_106(x))
Eq(F_106(x), F_107(x))
Eq(F_107(x), F_108(x)*F_12(x))
Eq(F_108(x), F_109(x) + F_112(x))
Eq(F_109(x), F_110(x))
Eq(F_110(x), F_111(x)*F_12(x))
Eq(F_111(x), F_109(x) + F_56(x))
Eq(F_112(x), F_107(x))
Eq(F_113(x), F_114(x) + F_116(x) + F_125(x) + 2*F_26(x))
Eq(F_114(x), F_115(x)*F_12(x))
Eq(F_115(x), F_41(x) + F_89(x))
Eq(F_116(x), F_117(x)*F_12(x))
Eq(F_117(x), F_118(x) + F_83(x))
Eq(F_118(x), F_113(x) + F_119(x))
Eq(F_119(x), F_120(x))
Eq(F_120(x), F_12(x)*F_121(x))
Eq(F_121(x), F_122(x) + F_127(x))
Eq(F_122(x), F_123(x) + F_125(x) + 3*F_26(x))
Eq(F_123(x), F_12(x)*F_124(x))
Eq(F_124(x), F_122(x) + F_84(x))
Eq(F_125(x), F_12(x)*F_126(x))
Eq(F_126(x), F_109(x) + F_122(x))
Eq(F_127(x), F_120(x))
Eq(F_128(x), F_12(x)*F_129(x))
Eq(F_129(x), F_130(x) + F_40(x))
Eq(F_130(x), F_131(x) + F_138(x))
Eq(F_131(x), F_132(x) + F_134(x) + F_26(x))
Eq(F_132(x), F_12(x)*F_133(x))
Eq(F_133(x), F_14(x) + F_52(x))
Eq(F_134(x), F_12(x)*F_135(x))
Eq(F_135(x), F_136(x) + F_20(x))
Eq(F_136(x), F_131(x) + F_137(x))
Eq(F_137(x), F_54(x))
Eq(F_138(x), F_139(x) + F_141(x) + F_26(x) + F_87(x))
Eq(F_139(x), F_12(x)*F_140(x))
Eq(F_140(x), F_25(x) + F_95(x))
Eq(F_141(x), F_12(x)*F_142(x))
Eq(F_142(x), F_143(x) + F_35(x))
Eq(F_143(x), F_138(x) + F_144(x))
Eq(F_144(x), F_82(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": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"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, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "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": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, 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