0213_1230_2301
Counting sequence:
1, 1, 2, 6, 21, 73, 238, 721, 2046, 5501, 14158, 35172, 84895, 200133, 462714, 1052727, 2363200, 5245849, 11535354, 25163458, 54518249, 117424881, 251631318, 536828413, 1140787234, 2415828037, 5100146838, 10737245792, 22548348403, 47244338829, 98783858242, 206157933027, 429496102116, 893352413505, 1855424900962, 3848289504766, 7971457847805, 16492672656985, 34084858344446, 70368741646761, 145135531856838, 299067159194701, 615726507362974, 1266637390285660, 2603643528841287, 5348024550840341, 10977524084002378, 22517998127954703, 46161896170318600, 94575592163059497, 193654783963542666, 396316767193355322, 810647932909372177, 1657324662852729217, 3386706919760456550, 6917529027616114645, 14123288431405807850, 28823037615139693461, 58798996734913963878, 119903836479072738904, 244419358976607706267, 498062089990109114461, 1014570924053971181970, 2066035336255409762747, 4205857648805711371180, 8559289250201158619985, 17413726405581735871858, 35417748621522250094454, 72016088863761991673637, 146393360968958894246505, 297509088420787530780430, 604462909807314458557057, 1227815285546107614792078, 2493409502955172519178269, 5062376869636259501578478, 10275869466724347802625364, 20853970388352353065348335, 42312403686512020899282277, 85833733192638671170394650, 174085318024506600904358887, 353006339327471718739939536, 715684085211860471129439161, 1450710983537555009326945306, 2940107593302778152539530674, 5957586439060892572579065913, 12069915383032457679864669969, 24449315775886260428825260278, 49517601571415210995499952045, 100273143182115802266329454514, 203022166442802365082920058853, 410996093042746251265934004534, 831895706399775544731594993232, 1683598453428117173862148778819, 3406810988113366516521683471661, 6892850138740997370637568395746, 13944156602510523416462928299923, 28205225855078104183300784774260, 57044277010270323067350725070433, 115356204620768875536199011717762, 233247710441994209875392345705710, 471566023284901337356772480749645
Generating function in Maple syntax:
(4*x^9-12*x^8+45*x^7-114*x^6+176*x^5-173*x^4+110*x^3-44*x^2+10*x-1)/(2*x-1)^2/(x-1)^7
Generating function in latex syntax:
\frac{4 x^{9}-12 x^{8}+45 x^{7}-114 x^{6}+176 x^{5}-173 x^{4}+110 x^{3}-44 x^{2}+10 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{7}}
Generating function in sympy syntax:
(4*x**9 - 12*x**8 + 45*x**7 - 114*x**6 + 176*x**5 - 173*x**4 + 110*x**3 - 44*x**2 + 10*x - 1)/((x - 1)**7*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^7*F(x)-4*x^9+12*x^8-45*x^7+114*x^6-176*x^5+173*x^4-110*x^3+44*x^2-10*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{7} F \! \left(x \right)-4 x^{9}+12 x^{8}-45 x^{7}+114 x^{6}-176 x^{5}+173 x^{4}-110 x^{3}+44 x^{2}-10 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,28+349/20*n-28*2^n-1/720*n^6+7/240*n^5-23/144*n^4+73/48*n^3+4*2^n*n+389/180*n^2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 28+\frac{349 n}{20}-28 \,2^{n}-\frac{n^{6}}{720}+\frac{7 n^{5}}{240}-\frac{23 n^{4}}{144}+\frac{73 n^{3}}{48}+4 \,2^{n} n +\frac{389 n^{2}}{180} & \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) = 238
a(7) = 721
a(8) = 2046
a(9) = 5501
a(n+2) = -1/720*n^6+11/240*n^5-65/144*n^4+43/16*n^3-2177/360*n^2+4*a(n+1)-4*a(n)+233/30*n-2, n >= 10
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) = 238
a \! \left(7\right) = 721
a \! \left(8\right) = 2046
a \! \left(9\right) = 5501
a \! \left(n +2\right) = -\frac{n^{6}}{720}+\frac{11 n^{5}}{240}-\frac{65 n^{4}}{144}+\frac{43 n^{3}}{16}-\frac{2177 n^{2}}{360}+4 a \! \left(n +1\right)-4 a \! \left(n \right)+\frac{233 n}{30}-2, \quad n \geq 10
Specification 1
Strategy pack name: insertion_row_and_col_placements
Tree: http://permpal.com/tree/329/
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[8,x]
F[4,x] = F[20,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[11,x]+F[6,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[14,x]+F[18,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[8,x]
F[17,x] = F[1,x]+F[15,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]
F[20,x] = F[21,x]+F[43,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[8,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[21,x]
F[26,x] = F[27,x]*F[8,x]
F[27,x] = F[17,x]+F[26,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[17,x]*F[30,x]*F[8,x]
F[30,x] = F[31,x]+F[38,x]
F[31,x] = F[17,x]+F[32,x]
F[32,x] = F[15,x]+F[33,x]
F[33,x] = F[34,x]+F[35,x]+F[37,x]
F[34,x] = 0
F[35,x] = F[36,x]*F[8,x]
F[36,x] = F[15,x]+F[33,x]
F[37,x] = F[32,x]*F[8,x]
F[38,x] = F[15,x]+F[39,x]
F[39,x] = F[34,x]+F[40,x]+F[42,x]
F[40,x] = F[41,x]*F[8,x]
F[41,x] = F[15,x]+F[39,x]
F[42,x] = F[38,x]*F[8,x]
F[43,x] = F[44,x]
F[44,x] = F[45,x]*F[8,x]
F[45,x] = F[46,x]+F[75,x]
F[46,x] = F[47,x]+F[68,x]
F[47,x] = F[48,x]+F[6,x]
F[48,x] = F[49,x]+F[6,x]
F[49,x] = F[34,x]+F[50,x]+F[62,x]
F[50,x] = F[51,x]*F[8,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[49,x]+F[6,x]
F[53,x] = F[11,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[56,x]*F[8,x]
F[56,x] = F[57,x]+F[61,x]
F[57,x] = F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[60,x]*F[8,x]
F[60,x] = F[15,x]+F[58,x]
F[61,x] = F[55,x]
F[62,x] = F[63,x]*F[8,x]
F[63,x] = F[14,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[34,x]+F[62,x]+F[66,x]
F[66,x] = F[67,x]*F[8,x]
F[67,x] = F[19,x]+F[6,x]
F[68,x] = F[69,x]
F[69,x] = F[17,x]*F[70,x]*F[8,x]
F[70,x] = F[6,x]+F[71,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[15,x]*F[17,x]
F[73,x] = F[74,x]
F[74,x] = F[17,x]*F[71,x]*F[8,x]
F[75,x] = F[168,x]+F[76,x]
F[76,x] = F[77,x]+F[83,x]
F[77,x] = F[6,x]*F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[8,x]*F[80,x]
F[80,x] = F[72,x]+F[81,x]
F[81,x] = F[1,x]+F[82,x]
F[82,x] = F[78,x]
F[83,x] = F[43,x]+F[84,x]
F[84,x] = F[34,x]+F[85,x]+F[90,x]
F[85,x] = F[8,x]*F[86,x]
F[86,x] = F[87,x]+F[88,x]
F[87,x] = F[49,x]+F[84,x]
F[88,x] = F[89,x]
F[89,x] = F[15,x]^3*F[17,x]
F[90,x] = F[91,x]
F[91,x] = F[8,x]*F[92,x]
F[92,x] = F[119,x]+F[93,x]
F[93,x] = F[94,x]+F[96,x]
F[94,x] = F[95,x]
F[95,x] = F[71,x]*F[78,x]
F[96,x] = F[15,x]*F[97,x]
F[97,x] = F[94,x]+F[98,x]
F[98,x] = F[78,x]+F[99,x]
F[99,x] = F[100,x]+F[34,x]+F[37,x]
F[100,x] = F[101,x]*F[8,x]
F[101,x] = F[102,x]+F[107,x]
F[102,x] = F[103,x]+F[15,x]
F[103,x] = F[104,x]+F[105,x]+F[34,x]
F[104,x] = F[15,x]*F[8,x]
F[105,x] = F[106,x]*F[8,x]
F[106,x] = F[103,x]+F[8,x]
F[107,x] = F[108,x]+F[99,x]
F[108,x] = F[109,x]+F[113,x]+F[114,x]+F[34,x]
F[109,x] = F[110,x]*F[8,x]
F[110,x] = F[111,x]+F[112,x]
F[111,x] = F[103,x]
F[112,x] = F[108,x]
F[113,x] = F[33,x]*F[8,x]
F[114,x] = F[115,x]*F[8,x]
F[115,x] = F[116,x]+F[118,x]
F[116,x] = F[117,x]
F[117,x] = F[15,x]*F[8,x]
F[118,x] = 2*F[34,x]+F[113,x]+F[114,x]
F[119,x] = F[120,x]+F[143,x]
F[120,x] = F[121,x]
F[121,x] = F[122,x]*F[71,x]
F[122,x] = F[123,x]+F[141,x]+F[34,x]
F[123,x] = F[124,x]*F[8,x]
F[124,x] = F[125,x]+F[131,x]
F[125,x] = F[126,x]+F[129,x]
F[126,x] = F[127,x]
F[127,x] = F[128,x]*F[8,x]
F[128,x] = F[1,x]+F[126,x]
F[129,x] = F[130,x]
F[130,x] = F[126,x]*F[8,x]
F[131,x] = F[122,x]+F[132,x]
F[132,x] = 2*F[34,x]+F[133,x]+F[137,x]
F[133,x] = F[134,x]*F[8,x]
F[134,x] = F[135,x]+F[136,x]
F[135,x] = F[129,x]
F[136,x] = F[132,x]
F[137,x] = F[138,x]*F[8,x]
F[138,x] = F[139,x]+F[141,x]+F[34,x]
F[139,x] = F[140,x]*F[8,x]
F[140,x] = F[126,x]+F[138,x]
F[141,x] = F[142,x]*F[8,x]
F[142,x] = F[138,x]+F[15,x]
F[143,x] = F[144,x]*F[15,x]
F[144,x] = F[120,x]+F[145,x]
F[145,x] = F[122,x]+F[146,x]
F[146,x] = F[114,x]+F[147,x]+F[165,x]+F[34,x]
F[147,x] = F[148,x]*F[8,x]
F[148,x] = F[149,x]+F[155,x]
F[149,x] = F[150,x]+F[153,x]
F[150,x] = F[105,x]+F[151,x]+F[34,x]
F[151,x] = F[152,x]*F[8,x]
F[152,x] = F[15,x]+F[150,x]
F[153,x] = F[154,x]
F[154,x] = F[150,x]*F[8,x]
F[155,x] = F[146,x]+F[156,x]
F[156,x] = 2*F[34,x]+F[157,x]+F[161,x]+F[167,x]
F[157,x] = F[158,x]*F[8,x]
F[158,x] = F[159,x]+F[160,x]
F[159,x] = F[153,x]
F[160,x] = F[156,x]
F[161,x] = F[162,x]*F[8,x]
F[162,x] = F[114,x]+F[163,x]+F[165,x]+F[34,x]
F[163,x] = F[164,x]*F[8,x]
F[164,x] = F[150,x]+F[162,x]
F[165,x] = F[166,x]*F[8,x]
F[166,x] = F[162,x]+F[33,x]
F[167,x] = 0
F[168,x] = F[169,x]+F[174,x]+F[34,x]
F[169,x] = F[170,x]*F[8,x]
F[170,x] = F[171,x]+F[172,x]
F[171,x] = F[168,x]+F[68,x]
F[172,x] = F[173,x]
F[173,x] = F[15,x]^2*F[17,x]*F[6,x]
F[174,x] = F[175,x]
F[175,x] = F[38,x]*F[70,x]*F[8,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{20}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{6}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \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_{21}\! \left(x \right)+F_{43}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{8}\! \left(x \right)
F_{27}\! \left(x \right) = F_{17}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{17}\! \left(x \right) F_{30}\! \left(x \right) F_{8}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{38}\! \left(x \right)
F_{31}\! \left(x \right) = F_{17}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = 0
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{8}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)
F_{37}\! \left(x \right) = F_{32}\! \left(x \right) F_{8}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{34}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right) F_{8}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right) F_{8}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right) F_{8}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{75}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{68}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{6}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{6}\! \left(x \right)
F_{49}\! \left(x \right) = F_{34}\! \left(x \right)+F_{50}\! \left(x \right)+F_{62}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right) F_{8}\! \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_{6}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right) F_{8}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right) F_{8}\! \left(x \right)
F_{60}\! \left(x \right) = F_{15}\! \left(x \right)+F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{55}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right) F_{8}\! \left(x \right)
F_{63}\! \left(x \right) = F_{14}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{34}\! \left(x \right)+F_{62}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right) F_{8}\! \left(x \right)
F_{67}\! \left(x \right) = F_{19}\! \left(x \right)+F_{6}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{17}\! \left(x \right) F_{70}\! \left(x \right) F_{8}\! \left(x \right)
F_{70}\! \left(x \right) = F_{6}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{73}\! \left(x \right)
F_{72}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{17}\! \left(x \right) F_{71}\! \left(x \right) F_{8}\! \left(x \right)
F_{75}\! \left(x \right) = F_{168}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{83}\! \left(x \right)
F_{77}\! \left(x \right) = F_{6}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{8}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{72}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{1}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{78}\! \left(x \right)
F_{83}\! \left(x \right) = F_{43}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{34}\! \left(x \right)+F_{85}\! \left(x \right)+F_{90}\! \left(x \right)
F_{85}\! \left(x \right) = F_{8}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{88}\! \left(x \right)
F_{87}\! \left(x \right) = F_{49}\! \left(x \right)+F_{84}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{15} \left(x \right)^{3} F_{17}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{8}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{119}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)+F_{96}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{71}\! \left(x \right) F_{78}\! \left(x \right)
F_{96}\! \left(x \right) = F_{15}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{94}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{78}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{8}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{107}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{15}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{105}\! \left(x \right)+F_{34}\! \left(x \right)
F_{104}\! \left(x \right) = F_{15}\! \left(x \right) F_{8}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{8}\! \left(x \right)
F_{106}\! \left(x \right) = F_{103}\! \left(x \right)+F_{8}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)+F_{99}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{113}\! \left(x \right)+F_{114}\! \left(x \right)+F_{34}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{8}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{112}\! \left(x \right)
F_{111}\! \left(x \right) = F_{103}\! \left(x \right)
F_{112}\! \left(x \right) = F_{108}\! \left(x \right)
F_{113}\! \left(x \right) = F_{33}\! \left(x \right) F_{8}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right) F_{8}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{118}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)
F_{117}\! \left(x \right) = F_{15}\! \left(x \right) F_{8}\! \left(x \right)
F_{118}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{113}\! \left(x \right)+F_{114}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)+F_{143}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right) F_{71}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)+F_{141}\! \left(x \right)+F_{34}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right) F_{8}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)+F_{131}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{129}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right) F_{8}\! \left(x \right)
F_{128}\! \left(x \right) = F_{1}\! \left(x \right)+F_{126}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = F_{126}\! \left(x \right) F_{8}\! \left(x \right)
F_{131}\! \left(x \right) = F_{122}\! \left(x \right)+F_{132}\! \left(x \right)
F_{132}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{133}\! \left(x \right)+F_{137}\! \left(x \right)
F_{133}\! \left(x \right) = F_{134}\! \left(x \right) F_{8}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)+F_{136}\! \left(x \right)
F_{135}\! \left(x \right) = F_{129}\! \left(x \right)
F_{136}\! \left(x \right) = F_{132}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right) F_{8}\! \left(x \right)
F_{138}\! \left(x \right) = F_{139}\! \left(x \right)+F_{141}\! \left(x \right)+F_{34}\! \left(x \right)
F_{139}\! \left(x \right) = F_{140}\! \left(x \right) F_{8}\! \left(x \right)
F_{140}\! \left(x \right) = F_{126}\! \left(x \right)+F_{138}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right) F_{8}\! \left(x \right)
F_{142}\! \left(x \right) = F_{138}\! \left(x \right)+F_{15}\! \left(x \right)
F_{143}\! \left(x \right) = F_{144}\! \left(x \right) F_{15}\! \left(x \right)
F_{144}\! \left(x \right) = F_{120}\! \left(x \right)+F_{145}\! \left(x \right)
F_{145}\! \left(x \right) = F_{122}\! \left(x \right)+F_{146}\! \left(x \right)
F_{146}\! \left(x \right) = F_{114}\! \left(x \right)+F_{147}\! \left(x \right)+F_{165}\! \left(x \right)+F_{34}\! \left(x \right)
F_{147}\! \left(x \right) = F_{148}\! \left(x \right) F_{8}\! \left(x \right)
F_{148}\! \left(x \right) = F_{149}\! \left(x \right)+F_{155}\! \left(x \right)
F_{149}\! \left(x \right) = F_{150}\! \left(x \right)+F_{153}\! \left(x \right)
F_{150}\! \left(x \right) = F_{105}\! \left(x \right)+F_{151}\! \left(x \right)+F_{34}\! \left(x \right)
F_{151}\! \left(x \right) = F_{152}\! \left(x \right) F_{8}\! \left(x \right)
F_{152}\! \left(x \right) = F_{15}\! \left(x \right)+F_{150}\! \left(x \right)
F_{153}\! \left(x \right) = F_{154}\! \left(x \right)
F_{154}\! \left(x \right) = F_{150}\! \left(x \right) F_{8}\! \left(x \right)
F_{155}\! \left(x \right) = F_{146}\! \left(x \right)+F_{156}\! \left(x \right)
F_{156}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{157}\! \left(x \right)+F_{161}\! \left(x \right)+F_{167}\! \left(x \right)
F_{157}\! \left(x \right) = F_{158}\! \left(x \right) F_{8}\! \left(x \right)
F_{158}\! \left(x \right) = F_{159}\! \left(x \right)+F_{160}\! \left(x \right)
F_{159}\! \left(x \right) = F_{153}\! \left(x \right)
F_{160}\! \left(x \right) = F_{156}\! \left(x \right)
F_{161}\! \left(x \right) = F_{162}\! \left(x \right) F_{8}\! \left(x \right)
F_{162}\! \left(x \right) = F_{114}\! \left(x \right)+F_{163}\! \left(x \right)+F_{165}\! \left(x \right)+F_{34}\! \left(x \right)
F_{163}\! \left(x \right) = F_{164}\! \left(x \right) F_{8}\! \left(x \right)
F_{164}\! \left(x \right) = F_{150}\! \left(x \right)+F_{162}\! \left(x \right)
F_{165}\! \left(x \right) = F_{166}\! \left(x \right) F_{8}\! \left(x \right)
F_{166}\! \left(x \right) = F_{162}\! \left(x \right)+F_{33}\! \left(x \right)
F_{167}\! \left(x \right) = 0
F_{168}\! \left(x \right) = F_{169}\! \left(x \right)+F_{174}\! \left(x \right)+F_{34}\! \left(x \right)
F_{169}\! \left(x \right) = F_{170}\! \left(x \right) F_{8}\! \left(x \right)
F_{170}\! \left(x \right) = F_{171}\! \left(x \right)+F_{172}\! \left(x \right)
F_{171}\! \left(x \right) = F_{168}\! \left(x \right)+F_{68}\! \left(x \right)
F_{172}\! \left(x \right) = F_{173}\! \left(x \right)
F_{173}\! \left(x \right) = F_{15} \left(x \right)^{2} F_{17}\! \left(x \right) F_{6}\! \left(x \right)
F_{174}\! \left(x \right) = F_{175}\! \left(x \right)
F_{175}\! \left(x \right) = F_{38}\! \left(x \right) F_{70}\! \left(x \right) F_{8}\! \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_8(x))
Eq(F_4(x), F_20(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_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_5(x))
Eq(F_10(x), F_11(x) + F_6(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_8(x))
Eq(F_13(x), F_14(x) + F_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_8(x))
Eq(F_17(x), F_1(x) + F_15(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x))
Eq(F_20(x), F_21(x) + F_43(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_8(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_21(x))
Eq(F_26(x), F_27(x)*F_8(x))
Eq(F_27(x), F_17(x) + F_26(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_17(x)*F_30(x)*F_8(x))
Eq(F_30(x), F_31(x) + F_38(x))
Eq(F_31(x), F_17(x) + F_32(x))
Eq(F_32(x), F_15(x) + F_33(x))
Eq(F_33(x), F_34(x) + F_35(x) + F_37(x))
Eq(F_34(x), 0)
Eq(F_35(x), F_36(x)*F_8(x))
Eq(F_36(x), F_15(x) + F_33(x))
Eq(F_37(x), F_32(x)*F_8(x))
Eq(F_38(x), F_15(x) + F_39(x))
Eq(F_39(x), F_34(x) + F_40(x) + F_42(x))
Eq(F_40(x), F_41(x)*F_8(x))
Eq(F_41(x), F_15(x) + F_39(x))
Eq(F_42(x), F_38(x)*F_8(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_45(x)*F_8(x))
Eq(F_45(x), F_46(x) + F_75(x))
Eq(F_46(x), F_47(x) + F_68(x))
Eq(F_47(x), F_48(x) + F_6(x))
Eq(F_48(x), F_49(x) + F_6(x))
Eq(F_49(x), F_34(x) + F_50(x) + F_62(x))
Eq(F_50(x), F_51(x)*F_8(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_49(x) + F_6(x))
Eq(F_53(x), F_11(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_56(x)*F_8(x))
Eq(F_56(x), F_57(x) + F_61(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_60(x)*F_8(x))
Eq(F_60(x), F_15(x) + F_58(x))
Eq(F_61(x), F_55(x))
Eq(F_62(x), F_63(x)*F_8(x))
Eq(F_63(x), F_14(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_34(x) + F_62(x) + F_66(x))
Eq(F_66(x), F_67(x)*F_8(x))
Eq(F_67(x), F_19(x) + F_6(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_17(x)*F_70(x)*F_8(x))
Eq(F_70(x), F_6(x) + F_71(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_15(x)*F_17(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_17(x)*F_71(x)*F_8(x))
Eq(F_75(x), F_168(x) + F_76(x))
Eq(F_76(x), F_77(x) + F_83(x))
Eq(F_77(x), F_6(x)*F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_8(x)*F_80(x))
Eq(F_80(x), F_72(x) + F_81(x))
Eq(F_81(x), F_1(x) + F_82(x))
Eq(F_82(x), F_78(x))
Eq(F_83(x), F_43(x) + F_84(x))
Eq(F_84(x), F_34(x) + F_85(x) + F_90(x))
Eq(F_85(x), F_8(x)*F_86(x))
Eq(F_86(x), F_87(x) + F_88(x))
Eq(F_87(x), F_49(x) + F_84(x))
Eq(F_88(x), F_89(x))
Eq(F_89(x), F_15(x)**3*F_17(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), F_8(x)*F_92(x))
Eq(F_92(x), F_119(x) + F_93(x))
Eq(F_93(x), F_94(x) + F_96(x))
Eq(F_94(x), F_95(x))
Eq(F_95(x), F_71(x)*F_78(x))
Eq(F_96(x), F_15(x)*F_97(x))
Eq(F_97(x), F_94(x) + F_98(x))
Eq(F_98(x), F_78(x) + F_99(x))
Eq(F_99(x), F_100(x) + F_34(x) + F_37(x))
Eq(F_100(x), F_101(x)*F_8(x))
Eq(F_101(x), F_102(x) + F_107(x))
Eq(F_102(x), F_103(x) + F_15(x))
Eq(F_103(x), F_104(x) + F_105(x) + F_34(x))
Eq(F_104(x), F_15(x)*F_8(x))
Eq(F_105(x), F_106(x)*F_8(x))
Eq(F_106(x), F_103(x) + F_8(x))
Eq(F_107(x), F_108(x) + F_99(x))
Eq(F_108(x), F_109(x) + F_113(x) + F_114(x) + F_34(x))
Eq(F_109(x), F_110(x)*F_8(x))
Eq(F_110(x), F_111(x) + F_112(x))
Eq(F_111(x), F_103(x))
Eq(F_112(x), F_108(x))
Eq(F_113(x), F_33(x)*F_8(x))
Eq(F_114(x), F_115(x)*F_8(x))
Eq(F_115(x), F_116(x) + F_118(x))
Eq(F_116(x), F_117(x))
Eq(F_117(x), F_15(x)*F_8(x))
Eq(F_118(x), F_113(x) + F_114(x) + 2*F_34(x))
Eq(F_119(x), F_120(x) + F_143(x))
Eq(F_120(x), F_121(x))
Eq(F_121(x), F_122(x)*F_71(x))
Eq(F_122(x), F_123(x) + F_141(x) + F_34(x))
Eq(F_123(x), F_124(x)*F_8(x))
Eq(F_124(x), F_125(x) + F_131(x))
Eq(F_125(x), F_126(x) + F_129(x))
Eq(F_126(x), F_127(x))
Eq(F_127(x), F_128(x)*F_8(x))
Eq(F_128(x), F_1(x) + F_126(x))
Eq(F_129(x), F_130(x))
Eq(F_130(x), F_126(x)*F_8(x))
Eq(F_131(x), F_122(x) + F_132(x))
Eq(F_132(x), F_133(x) + F_137(x) + 2*F_34(x))
Eq(F_133(x), F_134(x)*F_8(x))
Eq(F_134(x), F_135(x) + F_136(x))
Eq(F_135(x), F_129(x))
Eq(F_136(x), F_132(x))
Eq(F_137(x), F_138(x)*F_8(x))
Eq(F_138(x), F_139(x) + F_141(x) + F_34(x))
Eq(F_139(x), F_140(x)*F_8(x))
Eq(F_140(x), F_126(x) + F_138(x))
Eq(F_141(x), F_142(x)*F_8(x))
Eq(F_142(x), F_138(x) + F_15(x))
Eq(F_143(x), F_144(x)*F_15(x))
Eq(F_144(x), F_120(x) + F_145(x))
Eq(F_145(x), F_122(x) + F_146(x))
Eq(F_146(x), F_114(x) + F_147(x) + F_165(x) + F_34(x))
Eq(F_147(x), F_148(x)*F_8(x))
Eq(F_148(x), F_149(x) + F_155(x))
Eq(F_149(x), F_150(x) + F_153(x))
Eq(F_150(x), F_105(x) + F_151(x) + F_34(x))
Eq(F_151(x), F_152(x)*F_8(x))
Eq(F_152(x), F_15(x) + F_150(x))
Eq(F_153(x), F_154(x))
Eq(F_154(x), F_150(x)*F_8(x))
Eq(F_155(x), F_146(x) + F_156(x))
Eq(F_156(x), F_157(x) + F_161(x) + F_167(x) + 2*F_34(x))
Eq(F_157(x), F_158(x)*F_8(x))
Eq(F_158(x), F_159(x) + F_160(x))
Eq(F_159(x), F_153(x))
Eq(F_160(x), F_156(x))
Eq(F_161(x), F_162(x)*F_8(x))
Eq(F_162(x), F_114(x) + F_163(x) + F_165(x) + F_34(x))
Eq(F_163(x), F_164(x)*F_8(x))
Eq(F_164(x), F_150(x) + F_162(x))
Eq(F_165(x), F_166(x)*F_8(x))
Eq(F_166(x), F_162(x) + F_33(x))
Eq(F_167(x), 0)
Eq(F_168(x), F_169(x) + F_174(x) + F_34(x))
Eq(F_169(x), F_170(x)*F_8(x))
Eq(F_170(x), F_171(x) + F_172(x))
Eq(F_171(x), F_168(x) + F_68(x))
Eq(F_172(x), F_173(x))
Eq(F_173(x), F_15(x)**2*F_17(x)*F_6(x))
Eq(F_174(x), F_175(x))
Eq(F_175(x), F_38(x)*F_70(x)*F_8(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_row_and_col_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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