0132_1230_2130_3021
Counting sequence:
1, 1, 2, 6, 20, 61, 169, 442, 1120, 2789, 6877, 16858, 41172, 100301, 243905, 592298, 1436792, 3482373, 8434421, 20416874, 49399372, 119477901, 288879609, 698285850, 1687548624, 4077577573, 9851092557, 23796540090, 57477727364, 138819103885, 335250153073, 809587845706, 1954962715624, 4720587019013, 11398284237541, 27521450461642, 66449775095676, 160438180522445, 387360495879209, 935227891757882, 2257953718348736, 5451410206362597, 13161323886888125, 31775157491766938, 76713837893677876, 185207231325634125, 447137096637968673, 1079499016787616234, 2606170314585290328, 6291910014702374917, 15190131081478395861, 36672453652635877674, 88535601336703572908, 213744782225949866509, 516027417588416991577, 1245804121002411220570, 3007644666792494174128, 7261111468985909051237, 17529903633561331241005, 42320990793702609461626, 102172029336154626020580, 246665337696388013214989, 595503281189682955874513, 1437673052997258531811466, 3470851693027209233191880, 8379381050737695425583621, 20229623017874636939135429, 48838645533231043013406602, 117906950977824870385052380, 284652621275857078621718349, 687212341103491617304902537, 1659077598630745492584349818, 4005368128660792961179254432, 9669815036543952132354162661, 23345000562931938660710187165, 56359820884774312323419751258, 136064651777213529046840117652, 328489143328667301895680841933, 793042976213479995795363511681, 1914575171313491019400731285034, 4622193469956189486425472920632, 11158962413457324895908970803461, 26940118901333749085558001881269, 65039201425050642681654149272810, 157018524169286673678124649839884, 379076254599327268496420147777933, 915171043039347767587998343045817, 2209418360020835917506483629169050, 5334007801766645830269099191982224, 12877434040925380033380949194329445, 31088876038359910807703531943032333, 75055186427130211470133081805175866, 181199249511590353390659833002946884, 437453686688250957536833022710194573, 1056106625363972347035086428221585201, 2549666942367955808748526978749862602, 6155440520003404278815182584914304936, 14860548002181804994944976546964460805, 35876536563981095525837304475615202469, 86613621209372158560883923091738816842, 209103779141181737676133825846180737596
Generating function in Maple syntax:
-(2*x^7+5*x^6+2*x^5+x^4+2*x^3-8*x^2+5*x-1)/(2*x-1)/(x^2+2*x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{2 x^{7}+5 x^{6}+2 x^{5}+x^{4}+2 x^{3}-8 x^{2}+5 x -1}{\left(2 x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**7 - 5*x**6 - 2*x**5 - x**4 - 2*x**3 + 8*x**2 - 5*x + 1)/((x - 1)**2*(2*x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2+2*x-1)*(x-1)^2*F(x)+2*x^7+5*x^6+2*x^5+x^4+2*x^3-8*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+2 x^{7}+5 x^{6}+2 x^{5}+x^{4}+2 x^{3}-8 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/4*(-2^(1/2)+3)*(-1-2^(1/2))^(-n)+1/4*(2^(1/2)+3)*(2^(1/2)-1)^(-n)-4*n-2^(-1+n)+13/2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-\sqrt{2}+3\right) \left(-1-\sqrt{2}\right)^{-n}}{4}+\frac{\left(\sqrt{2}+3\right) \left(\sqrt{2}-1\right)^{-n}}{4}-4 n -2^{-1+n}+\frac{13}{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) = 20
a(5) = 61
a(6) = 169
a(7) = 442
a(n+3) = -2*a(n)-3*a(n+1)+4*a(n+2)-8*n+21, n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 20
a \! \left(5\right) = 61
a \! \left(6\right) = 169
a \! \left(7\right) = 442
a \! \left(n +3\right) = -2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)-8 n +21, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15641/
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[43,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[42,x]
F[42,x] = F[33,x]
F[43,x] = F[22,x]+F[44,x]+F[74,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[63,x]
F[46,x] = F[47,x]+F[6,x]
F[47,x] = F[22,x]+F[48,x]+F[53,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[10,x]+F[50,x]
F[50,x] = F[22,x]+F[48,x]+F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[37,x]+F[50,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[52,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[12,x]
F[59,x] = 2*F[22,x]+F[60,x]+F[62,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[26,x]
F[62,x] = F[12,x]*F[55,x]
F[63,x] = F[43,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[22,x]+F[68,x]+F[72,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[49,x]+F[70,x]
F[70,x] = F[67,x]+F[71,x]
F[71,x] = F[65,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[33,x]+F[67,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[73,x]+F[76,x]
F[76,x] = F[77,x]+F[80,x]
F[77,x] = F[78,x]
F[78,x] = F[12,x]*F[79,x]
F[79,x] = F[37,x]
F[80,x] = 2*F[22,x]+F[81,x]+F[83,x]
F[81,x] = F[12,x]*F[82,x]
F[82,x] = F[50,x]
F[83,x] = F[12,x]*F[76,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_{43}\! \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_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{33}\! \left(x \right)
F_{43}\! \left(x \right) = F_{22}\! \left(x \right)+F_{44}\! \left(x \right)+F_{74}\! \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_{63}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{6}\! \left(x \right)
F_{47}\! \left(x \right) = F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{53}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{37}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{52}\! \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_{12}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{26}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{63}\! \left(x \right) = F_{43}\! \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_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{22}\! \left(x \right)+F_{68}\! \left(x \right)+F_{72}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{49}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{67}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{65}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{33}\! \left(x \right)+F_{67}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{73}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{80}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{37}\! \left(x \right)
F_{80}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{81}\! \left(x \right)+F_{83}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{50}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \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_43(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_42(x))
Eq(F_42(x), F_33(x))
Eq(F_43(x), F_22(x) + F_44(x) + F_74(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_63(x))
Eq(F_46(x), F_47(x) + F_6(x))
Eq(F_47(x), F_22(x) + F_48(x) + F_53(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_10(x) + F_50(x))
Eq(F_50(x), F_22(x) + F_48(x) + F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_37(x) + F_50(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_52(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_12(x))
Eq(F_59(x), 2*F_22(x) + F_60(x) + F_62(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_26(x))
Eq(F_62(x), F_12(x)*F_55(x))
Eq(F_63(x), F_43(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_22(x) + F_68(x) + F_72(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_49(x) + F_70(x))
Eq(F_70(x), F_67(x) + F_71(x))
Eq(F_71(x), F_65(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_33(x) + F_67(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_73(x) + F_76(x))
Eq(F_76(x), F_77(x) + F_80(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_12(x)*F_79(x))
Eq(F_79(x), F_37(x))
Eq(F_80(x), 2*F_22(x) + F_81(x) + F_83(x))
Eq(F_81(x), F_12(x)*F_82(x))
Eq(F_82(x), F_50(x))
Eq(F_83(x), F_12(x)*F_76(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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