0123_0231_1203_3012_3021
Counting sequence:
1, 1, 2, 6, 19, 52, 126, 283, 607, 1266, 2596, 5269, 10629, 21364, 42850, 85839, 171835, 343846, 687888, 1375993, 2752225, 5504712, 11009710, 22019731, 44039799, 88079962, 176160316, 352321053, 704642557, 1409285596, 2818571706, 5637143959, 11274288499, 22548577614, 45097155880, 90194312449, 180388625625, 360777252016, 721554504838, 1443109010523, 2886218021935, 5772436044802, 11544872090580, 23089744182181, 46179488365429, 92358976731972, 184717953465106, 369435906931423, 738871813864107, 1477743627729526, 2955487255460416, 5910974510922249, 11821949021845969, 23643898043693464, 47287796087388510, 94575592174778659, 189151184349559015, 378302368699119786, 756604737398241388, 1513209474796484653, 3026418949592971245, 6052837899185944492, 12105675798371891050, 24211351596743784231, 48422703193487570659, 96845406386975143582, 193690812773950289496, 387381625547900581393, 774763251095801165257, 1549526502191602333056, 3099053004383204668726, 6198106008766409340139, 12396212017532818683039, 24792424035065637368914, 49584848070131274740740, 99169696140262549484469, 198339392280525098972005, 396678784561050197947156, 793357569122100395897538, 1586715138244200791798383, 3173430276488401583600155, 6346860552976803167203782, 12693721105953606334411120, 25387442211907212668825881, 50774884423814425337655489, 101549768847628850675314792, 203099537695257701350633486, 406199075390515402701270963, 812398150781030805402546007, 1624796301562061610805096186, 3249592603124123221610196636, 6499185206248246443220397629, 12998370412496492886440799709, 25996740824992985772881603964, 51993481649985971545763212570, 103986963299971943091526429879, 207973926599943886183052864595, 415947853199887772366105734126, 831895706399775544732211473288, 1663791412799551089464422951713, 3327582825599102178928845908665
Generating function in Maple syntax:
-(x^6+x^5-2*x^4+2*x^3-6*x^2+4*x-1)/(2*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{x^{6}+x^{5}-2 x^{4}+2 x^{3}-6 x^{2}+4 x -1}{\left(2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-x**6 - x**5 + 2*x**4 - 2*x**3 + 6*x**2 - 4*x + 1)/((x - 1)**3*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^3*F(x)+x^6+x^5-2*x^4+2*x^3-6*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{6}+x^{5}-2 x^{4}+2 x^{3}-6 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,3+21/8*2^n-9/2*n-1/2*n^2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 3+\frac{21 \,2^{n}}{8}-\frac{9 n}{2}-\frac{n^{2}}{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) = 19
a(5) = 52
a(6) = 126
a(n+1) = 1/2*n^2+2*a(n)+7/2*n-8, n >= 7
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) = 19
a \! \left(5\right) = 52
a \! \left(6\right) = 126
a \! \left(n +1\right) = \frac{n^{2}}{2}+2 a \! \left(n \right)+\frac{7 n}{2}-8, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17346/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[29,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[24,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[4,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[4,x]
F[23,x] = F[19,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[11,x]
F[28,x] = F[24,x]
F[29,x] = F[30,x]+F[40,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[17,x]+F[33,x]
F[33,x] = F[30,x]+F[34,x]
F[34,x] = F[35,x]+F[36,x]+F[39,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[22,x]+F[38,x]
F[38,x] = F[34,x]
F[39,x] = F[30,x]*F[4,x]
F[40,x] = F[35,x]+F[41,x]+F[78,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[56,x]
F[43,x] = F[44,x]+F[7,x]
F[44,x] = F[35,x]+F[45,x]+F[46,x]
F[45,x] = F[11,x]*F[4,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[4,x]
F[49,x] = F[50,x]
F[50,x] = F[35,x]+F[51,x]+F[52,x]
F[51,x] = x^2
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[4,x]
F[55,x] = F[50,x]
F[56,x] = F[40,x]+F[57,x]
F[57,x] = F[35,x]+F[58,x]+F[62,x]+F[66,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[44,x]
F[61,x] = F[57,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[30,x]+F[63,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]+F[69,x]
F[68,x] = F[34,x]
F[69,x] = F[70,x]
F[70,x] = F[35,x]+F[71,x]+F[72,x]+F[74,x]
F[71,x] = 0
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[39,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]+F[77,x]
F[76,x] = F[34,x]
F[77,x] = F[70,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[65,x]+F[80,x]
F[80,x] = F[81,x]+F[91,x]
F[81,x] = F[35,x]+F[36,x]+F[82,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[84,x]+F[85,x]
F[84,x] = F[30,x]+F[73,x]
F[85,x] = F[81,x]+F[86,x]
F[86,x] = F[87,x]
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[89,x]+F[90,x]
F[89,x] = F[73,x]
F[90,x] = F[86,x]
F[91,x] = F[92,x]
F[92,x] = F[4,x]*F[93,x]
F[93,x] = F[94,x]+F[95,x]
F[94,x] = F[63,x]
F[95,x] = F[91,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{29}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{24}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \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_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)
F_{28}\! \left(x \right) = F_{24}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{40}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{17}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{22}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right)
F_{39}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{35}\! \left(x \right)+F_{41}\! \left(x \right)+F_{78}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{56}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{7}\! \left(x \right)
F_{44}\! \left(x \right) = F_{35}\! \left(x \right)+F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{35}\! \left(x \right)+F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = x^{2}
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right)
F_{55}\! \left(x \right) = F_{50}\! \left(x \right)
F_{56}\! \left(x \right) = F_{40}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{35}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)+F_{66}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{44}\! \left(x \right)
F_{61}\! \left(x \right) = F_{57}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{30}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{69}\! \left(x \right)
F_{68}\! \left(x \right) = F_{34}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{35}\! \left(x \right)+F_{71}\! \left(x \right)+F_{72}\! \left(x \right)+F_{74}\! \left(x \right)
F_{71}\! \left(x \right) = 0
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{39}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{76}\! \left(x \right) = F_{34}\! \left(x \right)
F_{77}\! \left(x \right) = F_{70}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{65}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{91}\! \left(x \right)
F_{81}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{85}\! \left(x \right)
F_{84}\! \left(x \right) = F_{30}\! \left(x \right)+F_{73}\! \left(x \right)
F_{85}\! \left(x \right) = F_{81}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{90}\! \left(x \right)
F_{89}\! \left(x \right) = F_{73}\! \left(x \right)
F_{90}\! \left(x \right) = F_{86}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)+F_{95}\! \left(x \right)
F_{94}\! \left(x \right) = F_{63}\! \left(x \right)
F_{95}\! \left(x \right) = F_{91}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_29(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_24(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_4(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_4(x))
Eq(F_23(x), F_19(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_11(x))
Eq(F_28(x), F_24(x))
Eq(F_29(x), F_30(x) + F_40(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_17(x) + F_33(x))
Eq(F_33(x), F_30(x) + F_34(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_39(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_22(x) + F_38(x))
Eq(F_38(x), F_34(x))
Eq(F_39(x), F_30(x)*F_4(x))
Eq(F_40(x), F_35(x) + F_41(x) + F_78(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_56(x))
Eq(F_43(x), F_44(x) + F_7(x))
Eq(F_44(x), F_35(x) + F_45(x) + F_46(x))
Eq(F_45(x), F_11(x)*F_4(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_4(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_35(x) + F_51(x) + F_52(x))
Eq(F_51(x), x**2)
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_4(x))
Eq(F_55(x), F_50(x))
Eq(F_56(x), F_40(x) + F_57(x))
Eq(F_57(x), F_35(x) + F_58(x) + F_62(x) + F_66(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_44(x))
Eq(F_61(x), F_57(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_30(x) + F_63(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_69(x))
Eq(F_68(x), F_34(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_35(x) + F_71(x) + F_72(x) + F_74(x))
Eq(F_71(x), 0)
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_39(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_77(x))
Eq(F_76(x), F_34(x))
Eq(F_77(x), F_70(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_65(x) + F_80(x))
Eq(F_80(x), F_81(x) + F_91(x))
Eq(F_81(x), F_35(x) + F_36(x) + F_82(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_84(x) + F_85(x))
Eq(F_84(x), F_30(x) + F_73(x))
Eq(F_85(x), F_81(x) + F_86(x))
Eq(F_86(x), F_87(x))
Eq(F_87(x), F_4(x)*F_88(x))
Eq(F_88(x), F_89(x) + F_90(x))
Eq(F_89(x), F_73(x))
Eq(F_90(x), F_86(x))
Eq(F_91(x), F_92(x))
Eq(F_92(x), F_4(x)*F_93(x))
Eq(F_93(x), F_94(x) + F_95(x))
Eq(F_94(x), F_63(x))
Eq(F_95(x), F_91(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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