0132_1230_2130
Counting sequence:
1, 1, 2, 6, 21, 75, 262, 890, 2949, 9575, 30590, 96486, 301269, 933171, 2872102, 8794946, 26822901, 81539855, 247232702, 748061070, 2259653349, 6816525435, 20540701510, 61842968906, 186063857829, 559486534391, 1681592864702, 5052356855286, 15175393904181, 45570473759235, 136818462927142, 410714032562450, 1232766963769749, 3699810340996895, 11103076935932030, 33318036378333726, 99975374778952389, 299977478375166411, 900056443205807302, 2700468768535824026, 8102129328963537861, 24308133747399054215, 72928616336325923582, 218796026057255361030, 656412649561473280149, 1969297272910158335955, 5908035047367752218150, 17724450941195735614946, 53174187685593814064949, 159524578650255879773231, 478578602140460610051902, 1435747554675717185909166, 4307271027288465193387941, 12921881557768258114080795, 38765809990623612859422022, 116297829086914977773912810, 348894450817159249484952549, 1046685678697364918486414615, 3140062652176312208704535870, 9420201515015316296673676566, 28260637278247040872462315509, 84781990859917916628981298851, 254346163363884890644877310502, 763038950686246668966699958706, 2289117964034360984731383038421, 6867356576653528592907210854975, 20602076211046321406015893982462, 61806244279879327855422976073790, 185418770614241339956461196994949, 556256403038744883263438260864875, 1668769429283027306921206067541190, 5006308819378983246324634795222586, 15018927741364135343037187561464069, 45056786322077859154419863039887271, 135170366445434030501180907829055102, 405511117392693173071176530906660646, 1216533395770071580119684265558793301, 3649600292550608833204927428354352755, 10948801131724644525144549178419524902, 32846404008560039277202781867972899970, 98539213506525298376403832991350655349, 295617644094652664152026511738923878735, 886852940914957315509414855448534770110, 2660558843581948570583549834203177197006, 7981676581051000700766234587814789910629, 23945029864600393540088457660229158082491, 71835089887001128156267022776495706917702, 215505270368850690321407235651955169742986, 646515812815446668891061919064743430476581, 1939547442571976585750724804070721364110135, 5818642337676097668076597628746558521329342, 17455927037074265714441188187530024219447350, 52367781169274911093540798024395301847821749, 157103343647974935129208295583466987477008195, 471310031282277325510894001803528905285838630, 1413930094663687223579566294168644783097183826, 4241790285963124694859256889358220625818695189, 12725370862650355399672811599103376896136028511, 38176112599445091892943155923352015799067090302, 114528337826084308461002113467432870230743173726, 343585013545245016799732677951622816657060232261
Generating function in Maple syntax:
(2*x^5-4*x^4-10*x^3+16*x^2-7*x+1)/(x-1)/(3*x-1)/(2*x-1)/(x^2+2*x-1)
Generating function in latex syntax:
\frac{2 x^{5}-4 x^{4}-10 x^{3}+16 x^{2}-7 x +1}{\left(x -1\right) \left(3 x -1\right) \left(2 x -1\right) \left(x^{2}+2 x -1\right)}
Generating function in sympy syntax:
(2*x**5 - 4*x**4 - 10*x**3 + 16*x**2 - 7*x + 1)/((x - 1)*(2*x - 1)*(3*x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(3*x-1)*(2*x-1)*(x^2+2*x-1)*F(x)-2*x^5+4*x^4+10*x^3-16*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(3 x -1\right) \left(2 x -1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)-2 x^{5}+4 x^{4}+10 x^{3}-16 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/4*(2*2^(1/2)-3)*(-1-2^(1/2))^(-n)+1/4*(-2*2^(1/2)-3)*(2^(1/2)-1)^(-n)+2^n+2*3^(n-1)+1/2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(2 \sqrt{2}-3\right) \left(-1-\sqrt{2}\right)^{-n}}{4}+\frac{\left(-2 \sqrt{2}-3\right) \left(\sqrt{2}-1\right)^{-n}}{4}+2^{n}+2 \,3^{n -1}+\\\frac{1}{2} & \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) = 75
a(n+4) = 6*a(n)+7*a(n+1)-15*a(n+2)+7*a(n+3)-2, n >= 6
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) = 75
a \! \left(n +4\right) = 6 a \! \left(n \right)+7 a \! \left(n +1\right)-15 a \! \left(n +2\right)+7 a \! \left(n +3\right)-2, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14726/
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[32,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[21,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[14,x]
F[21,x] = F[22,x]+F[23,x]+F[31,x]
F[22,x] = 0
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[10,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[10,x]*F[12,x]
F[28,x] = F[21,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[21,x]
F[31,x] = F[12,x]*F[13,x]
F[32,x] = F[33,x]+F[42,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[1,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[36,x]
F[39,x] = F[33,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[39,x]
F[42,x] = F[120,x]+F[22,x]+F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[78,x]
F[45,x] = F[46,x]+F[6,x]
F[46,x] = F[22,x]+F[47,x]+F[52,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[10,x]+F[49,x]
F[49,x] = F[22,x]+F[47,x]+F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[37,x]+F[49,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[51,x]+F[54,x]
F[54,x] = F[55,x]+F[66,x]
F[55,x] = F[22,x]+F[56,x]+F[60,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[12,x]+F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[12,x]*F[37,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[37,x]+F[58,x]
F[63,x] = F[55,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[12,x]*F[55,x]
F[66,x] = F[22,x]+F[67,x]+F[71,x]+F[77,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[26,x]+F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[12,x]*F[49,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[73,x]+F[74,x]
F[73,x] = F[49,x]+F[69,x]
F[74,x] = F[66,x]+F[75,x]
F[75,x] = F[76,x]
F[76,x] = F[12,x]*F[66,x]
F[77,x] = F[12,x]*F[54,x]
F[78,x] = F[42,x]+F[79,x]
F[79,x] = 2*F[22,x]+F[80,x]+F[90,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[82,x]+F[87,x]
F[82,x] = F[22,x]+F[83,x]+F[85,x]
F[83,x] = F[12,x]*F[84,x]
F[84,x] = F[48,x]+F[81,x]
F[85,x] = F[12,x]*F[86,x]
F[86,x] = F[33,x]+F[82,x]
F[87,x] = 2*F[22,x]+F[80,x]+F[88,x]
F[88,x] = F[12,x]*F[89,x]
F[89,x] = F[40,x]+F[87,x]
F[90,x] = F[12,x]*F[91,x]
F[91,x] = F[89,x]+F[92,x]
F[92,x] = F[106,x]+F[93,x]
F[93,x] = 2*F[22,x]+F[100,x]+F[94,x]
F[94,x] = F[12,x]*F[95,x]
F[95,x] = F[96,x]+F[98,x]
F[96,x] = F[97,x]
F[97,x] = F[12,x]*F[33,x]
F[98,x] = F[99,x]
F[99,x] = F[12,x]*F[40,x]
F[100,x] = F[101,x]*F[12,x]
F[101,x] = F[102,x]+F[103,x]
F[102,x] = F[40,x]+F[98,x]
F[103,x] = F[104,x]+F[93,x]
F[104,x] = F[105,x]
F[105,x] = F[12,x]*F[93,x]
F[106,x] = 2*F[22,x]+F[107,x]+F[113,x]+F[119,x]
F[107,x] = F[108,x]*F[12,x]
F[108,x] = F[109,x]+F[111,x]
F[109,x] = F[110,x]
F[110,x] = F[12,x]*F[82,x]
F[111,x] = F[112,x]
F[112,x] = F[12,x]*F[87,x]
F[113,x] = F[114,x]*F[12,x]
F[114,x] = F[115,x]+F[116,x]
F[115,x] = F[111,x]+F[87,x]
F[116,x] = F[106,x]+F[117,x]
F[117,x] = F[118,x]
F[118,x] = F[106,x]*F[12,x]
F[119,x] = F[12,x]*F[92,x]
F[120,x] = F[12,x]*F[121,x]
F[121,x] = F[122,x]+F[86,x]
F[122,x] = F[123,x]+F[140,x]
F[123,x] = F[124,x]+F[134,x]+F[22,x]
F[124,x] = F[12,x]*F[125,x]
F[125,x] = F[126,x]+F[130,x]
F[126,x] = F[127,x]+F[37,x]
F[127,x] = F[128,x]
F[128,x] = F[12,x]*F[129,x]
F[129,x] = F[127,x]+F[37,x]
F[130,x] = F[131,x]+F[40,x]
F[131,x] = F[132,x]
F[132,x] = F[12,x]*F[133,x]
F[133,x] = F[131,x]+F[40,x]
F[134,x] = F[12,x]*F[135,x]
F[135,x] = F[136,x]+F[137,x]
F[136,x] = F[33,x]+F[96,x]
F[137,x] = F[123,x]+F[138,x]
F[138,x] = F[139,x]
F[139,x] = F[12,x]*F[123,x]
F[140,x] = F[141,x]+F[155,x]+F[161,x]+F[22,x]
F[141,x] = F[12,x]*F[142,x]
F[142,x] = F[143,x]+F[149,x]
F[143,x] = F[144,x]+F[49,x]
F[144,x] = 2*F[22,x]+F[145,x]+F[147,x]
F[145,x] = F[12,x]*F[146,x]
F[146,x] = F[144,x]+F[49,x]
F[147,x] = F[12,x]*F[148,x]
F[148,x] = F[127,x]+F[144,x]
F[149,x] = F[150,x]+F[87,x]
F[150,x] = 3*F[22,x]+F[151,x]+F[153,x]
F[151,x] = F[12,x]*F[152,x]
F[152,x] = F[150,x]+F[87,x]
F[153,x] = F[12,x]*F[154,x]
F[154,x] = F[131,x]+F[150,x]
F[155,x] = F[12,x]*F[156,x]
F[156,x] = F[157,x]+F[158,x]
F[157,x] = F[109,x]+F[82,x]
F[158,x] = F[140,x]+F[159,x]
F[159,x] = F[160,x]
F[160,x] = F[12,x]*F[140,x]
F[161,x] = F[12,x]*F[122,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_{32}\! \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_{21}\! \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_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{31}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{28}\! \left(x \right) = F_{21}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{42}\! \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_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{120}\! \left(x \right)+F_{22}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{78}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{6}\! \left(x \right)
F_{46}\! \left(x \right) = F_{22}\! \left(x \right)+F_{47}\! \left(x \right)+F_{52}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{22}\! \left(x \right)+F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{37}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{66}\! \left(x \right)
F_{55}\! \left(x \right) = F_{22}\! \left(x \right)+F_{56}\! \left(x \right)+F_{60}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \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_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{37}\! \left(x \right)+F_{58}\! \left(x \right)
F_{63}\! \left(x \right) = F_{55}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{66}\! \left(x \right) = F_{22}\! \left(x \right)+F_{67}\! \left(x \right)+F_{71}\! \left(x \right)+F_{77}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{26}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{74}\! \left(x \right)
F_{73}\! \left(x \right) = F_{49}\! \left(x \right)+F_{69}\! \left(x \right)
F_{74}\! \left(x \right) = F_{66}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{78}\! \left(x \right) = F_{42}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{80}\! \left(x \right)+F_{90}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{87}\! \left(x \right)
F_{82}\! \left(x \right) = F_{22}\! \left(x \right)+F_{83}\! \left(x \right)+F_{85}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{48}\! \left(x \right)+F_{81}\! \left(x \right)
F_{85}\! \left(x \right) = F_{12}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{33}\! \left(x \right)+F_{82}\! \left(x \right)
F_{87}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{80}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{12}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{40}\! \left(x \right)+F_{87}\! \left(x \right)
F_{90}\! \left(x \right) = F_{12}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{89}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{106}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{100}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{12}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)+F_{98}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{98}\! \left(x \right) = F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{12}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{103}\! \left(x \right)
F_{102}\! \left(x \right) = F_{40}\! \left(x \right)+F_{98}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{93}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)
F_{105}\! \left(x \right) = F_{12}\! \left(x \right) F_{93}\! \left(x \right)
F_{106}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{107}\! \left(x \right)+F_{113}\! \left(x \right)+F_{119}\! \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_{111}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)
F_{110}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{12}\! \left(x \right) F_{87}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right) F_{12}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{116}\! \left(x \right)
F_{115}\! \left(x \right) = F_{111}\! \left(x \right)+F_{87}\! \left(x \right)
F_{116}\! \left(x \right) = F_{106}\! \left(x \right)+F_{117}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)
F_{118}\! \left(x \right) = F_{106}\! \left(x \right) F_{12}\! \left(x \right)
F_{119}\! \left(x \right) = F_{12}\! \left(x \right) F_{92}\! \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_{86}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)+F_{140}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)+F_{134}\! \left(x \right)+F_{22}\! \left(x \right)
F_{124}\! \left(x \right) = F_{12}\! \left(x \right) F_{125}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{130}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{37}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)
F_{128}\! \left(x \right) = F_{12}\! \left(x \right) F_{129}\! \left(x \right)
F_{129}\! \left(x \right) = F_{127}\! \left(x \right)+F_{37}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right)+F_{40}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)
F_{132}\! \left(x \right) = F_{12}\! \left(x \right) F_{133}\! \left(x \right)
F_{133}\! \left(x \right) = F_{131}\! \left(x \right)+F_{40}\! \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_{137}\! \left(x \right)
F_{136}\! \left(x \right) = F_{33}\! \left(x \right)+F_{96}\! \left(x \right)
F_{137}\! \left(x \right) = F_{123}\! \left(x \right)+F_{138}\! \left(x \right)
F_{138}\! \left(x \right) = F_{139}\! \left(x \right)
F_{139}\! \left(x \right) = F_{12}\! \left(x \right) F_{123}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right)+F_{155}\! \left(x \right)+F_{161}\! \left(x \right)+F_{22}\! \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_{149}\! \left(x \right)
F_{143}\! \left(x \right) = F_{144}\! \left(x \right)+F_{49}\! \left(x \right)
F_{144}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{145}\! \left(x \right)+F_{147}\! \left(x \right)
F_{145}\! \left(x \right) = F_{12}\! \left(x \right) F_{146}\! \left(x \right)
F_{146}\! \left(x \right) = F_{144}\! \left(x \right)+F_{49}\! \left(x \right)
F_{147}\! \left(x \right) = F_{12}\! \left(x \right) F_{148}\! \left(x \right)
F_{148}\! \left(x \right) = F_{127}\! \left(x \right)+F_{144}\! \left(x \right)
F_{149}\! \left(x \right) = F_{150}\! \left(x \right)+F_{87}\! \left(x \right)
F_{150}\! \left(x \right) = 3 F_{22}\! \left(x \right)+F_{151}\! \left(x \right)+F_{153}\! \left(x \right)
F_{151}\! \left(x \right) = F_{12}\! \left(x \right) F_{152}\! \left(x \right)
F_{152}\! \left(x \right) = F_{150}\! \left(x \right)+F_{87}\! \left(x \right)
F_{153}\! \left(x \right) = F_{12}\! \left(x \right) F_{154}\! \left(x \right)
F_{154}\! \left(x \right) = F_{131}\! \left(x \right)+F_{150}\! \left(x \right)
F_{155}\! \left(x \right) = F_{12}\! \left(x \right) F_{156}\! \left(x \right)
F_{156}\! \left(x \right) = F_{157}\! \left(x \right)+F_{158}\! \left(x \right)
F_{157}\! \left(x \right) = F_{109}\! \left(x \right)+F_{82}\! \left(x \right)
F_{158}\! \left(x \right) = F_{140}\! \left(x \right)+F_{159}\! \left(x \right)
F_{159}\! \left(x \right) = F_{160}\! \left(x \right)
F_{160}\! \left(x \right) = F_{12}\! \left(x \right) F_{140}\! \left(x \right)
F_{161}\! \left(x \right) = F_{12}\! \left(x \right) F_{122}\! \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_32(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_21(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_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_14(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_31(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_10(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_10(x)*F_12(x))
Eq(F_28(x), F_21(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_21(x))
Eq(F_31(x), F_12(x)*F_13(x))
Eq(F_32(x), F_33(x) + F_42(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_39(x))
Eq(F_36(x), F_1(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_36(x))
Eq(F_39(x), F_33(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_39(x))
Eq(F_42(x), F_120(x) + F_22(x) + F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_78(x))
Eq(F_45(x), F_46(x) + F_6(x))
Eq(F_46(x), F_22(x) + F_47(x) + F_52(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_10(x) + F_49(x))
Eq(F_49(x), F_22(x) + F_47(x) + F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_37(x) + F_49(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_51(x) + F_54(x))
Eq(F_54(x), F_55(x) + F_66(x))
Eq(F_55(x), F_22(x) + F_56(x) + F_60(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_12(x) + F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_12(x)*F_37(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_37(x) + F_58(x))
Eq(F_63(x), F_55(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_12(x)*F_55(x))
Eq(F_66(x), F_22(x) + F_67(x) + F_71(x) + F_77(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_26(x) + F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_12(x)*F_49(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_74(x))
Eq(F_73(x), F_49(x) + F_69(x))
Eq(F_74(x), F_66(x) + F_75(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_12(x)*F_66(x))
Eq(F_77(x), F_12(x)*F_54(x))
Eq(F_78(x), F_42(x) + F_79(x))
Eq(F_79(x), 2*F_22(x) + F_80(x) + F_90(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_82(x) + F_87(x))
Eq(F_82(x), F_22(x) + F_83(x) + F_85(x))
Eq(F_83(x), F_12(x)*F_84(x))
Eq(F_84(x), F_48(x) + F_81(x))
Eq(F_85(x), F_12(x)*F_86(x))
Eq(F_86(x), F_33(x) + F_82(x))
Eq(F_87(x), 2*F_22(x) + F_80(x) + F_88(x))
Eq(F_88(x), F_12(x)*F_89(x))
Eq(F_89(x), F_40(x) + F_87(x))
Eq(F_90(x), F_12(x)*F_91(x))
Eq(F_91(x), F_89(x) + F_92(x))
Eq(F_92(x), F_106(x) + F_93(x))
Eq(F_93(x), F_100(x) + 2*F_22(x) + F_94(x))
Eq(F_94(x), F_12(x)*F_95(x))
Eq(F_95(x), F_96(x) + F_98(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), F_12(x)*F_33(x))
Eq(F_98(x), F_99(x))
Eq(F_99(x), F_12(x)*F_40(x))
Eq(F_100(x), F_101(x)*F_12(x))
Eq(F_101(x), F_102(x) + F_103(x))
Eq(F_102(x), F_40(x) + F_98(x))
Eq(F_103(x), F_104(x) + F_93(x))
Eq(F_104(x), F_105(x))
Eq(F_105(x), F_12(x)*F_93(x))
Eq(F_106(x), F_107(x) + F_113(x) + F_119(x) + 2*F_22(x))
Eq(F_107(x), F_108(x)*F_12(x))
Eq(F_108(x), F_109(x) + F_111(x))
Eq(F_109(x), F_110(x))
Eq(F_110(x), F_12(x)*F_82(x))
Eq(F_111(x), F_112(x))
Eq(F_112(x), F_12(x)*F_87(x))
Eq(F_113(x), F_114(x)*F_12(x))
Eq(F_114(x), F_115(x) + F_116(x))
Eq(F_115(x), F_111(x) + F_87(x))
Eq(F_116(x), F_106(x) + F_117(x))
Eq(F_117(x), F_118(x))
Eq(F_118(x), F_106(x)*F_12(x))
Eq(F_119(x), F_12(x)*F_92(x))
Eq(F_120(x), F_12(x)*F_121(x))
Eq(F_121(x), F_122(x) + F_86(x))
Eq(F_122(x), F_123(x) + F_140(x))
Eq(F_123(x), F_124(x) + F_134(x) + F_22(x))
Eq(F_124(x), F_12(x)*F_125(x))
Eq(F_125(x), F_126(x) + F_130(x))
Eq(F_126(x), F_127(x) + F_37(x))
Eq(F_127(x), F_128(x))
Eq(F_128(x), F_12(x)*F_129(x))
Eq(F_129(x), F_127(x) + F_37(x))
Eq(F_130(x), F_131(x) + F_40(x))
Eq(F_131(x), F_132(x))
Eq(F_132(x), F_12(x)*F_133(x))
Eq(F_133(x), F_131(x) + F_40(x))
Eq(F_134(x), F_12(x)*F_135(x))
Eq(F_135(x), F_136(x) + F_137(x))
Eq(F_136(x), F_33(x) + F_96(x))
Eq(F_137(x), F_123(x) + F_138(x))
Eq(F_138(x), F_139(x))
Eq(F_139(x), F_12(x)*F_123(x))
Eq(F_140(x), F_141(x) + F_155(x) + F_161(x) + F_22(x))
Eq(F_141(x), F_12(x)*F_142(x))
Eq(F_142(x), F_143(x) + F_149(x))
Eq(F_143(x), F_144(x) + F_49(x))
Eq(F_144(x), F_145(x) + F_147(x) + 2*F_22(x))
Eq(F_145(x), F_12(x)*F_146(x))
Eq(F_146(x), F_144(x) + F_49(x))
Eq(F_147(x), F_12(x)*F_148(x))
Eq(F_148(x), F_127(x) + F_144(x))
Eq(F_149(x), F_150(x) + F_87(x))
Eq(F_150(x), F_151(x) + F_153(x) + 3*F_22(x))
Eq(F_151(x), F_12(x)*F_152(x))
Eq(F_152(x), F_150(x) + F_87(x))
Eq(F_153(x), F_12(x)*F_154(x))
Eq(F_154(x), F_131(x) + F_150(x))
Eq(F_155(x), F_12(x)*F_156(x))
Eq(F_156(x), F_157(x) + F_158(x))
Eq(F_157(x), F_109(x) + F_82(x))
Eq(F_158(x), F_140(x) + F_159(x))
Eq(F_159(x), F_160(x))
Eq(F_160(x), F_12(x)*F_140(x))
Eq(F_161(x), F_12(x)*F_122(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, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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": 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