0132_0213_1302_2031_3120
Counting sequence:
1, 1, 2, 6, 19, 53, 134, 319, 731, 1631, 3567, 7684, 16371, 34613, 72816, 152705, 319623, 668143, 1395329, 2911342, 6068831, 12638085, 26290090, 54628163, 113382275, 235059823, 486771443, 1006931192, 2080754043, 4295445157, 8858967012, 18254350853, 37581873999, 77310523295, 158915139717, 326419143010, 670016851847, 1374391865941, 2817501313886, 5772439316103, 11819753845291, 24189260316479, 49478028505463, 101155075862380, 206708193092291, 422212473223253, 862017125556312, 1759218615193225, 3588805965347415, 7318349408476943, 14918173781572617, 30399297502778582, 61924494896722287, 126100789589352037, 256705178785967378, 522417556803990475, 1062849512091930131, 2161727821174152399, 4395513236354107003, 8935141660748150624, 18158513697607934411, 36893488147474660933, 74939897799506561996, 152185638608171782157, 308982963234710001567, 627189298506207393343, 1272825341086049961741, 2582544170319437065418, 5238875316933622157847, 10625324586456821655989, 21545797078092887453382, 43681889966544361497871, 88544371553806004050235, 179449926349046688407839, 363622219180962866764799, 736689171327664854758100, 1492267808586808106211475, 3022314549036573173926517, 6120186961799060453871680, 12391489651049949318773905, 25085210757003555675729191, 50774884423814425662278447, 102758694667243480200268689, 207935240973716218426994494, 420706185225890953204319103, 851083777008698939430593093, 1721510367131231945251844730, 3481706360490132023658866771, 7040783973435600314030802339, 14236310451781873161921141295, 28782105913385091392027362307, 58183181846412872920925515720, 117604303732111126116129986843, 237684487542793012781394217701, 480320735242727546661674540980, 970544990799738135521782628885, 1960897022228042355441139952879, 3961408125713216879678185827295, 8002044413940698096948991744917, 16162545152909924869084086544946, 32642002955876907088541299742631
Generating function in Maple syntax:
(5*x^9-29*x^8+69*x^7-124*x^6+174*x^5-171*x^4+110*x^3-44*x^2+10*x-1)/(2*x-1)^2/(-1+x)^7
Generating function in latex syntax:
\frac{5 x^{9}-29 x^{8}+69 x^{7}-124 x^{6}+174 x^{5}-171 x^{4}+110 x^{3}-44 x^{2}+10 x -1}{\left(2 x -1\right)^{2} \left(-1+x \right)^{7}}
Generating function in sympy syntax:
(5*x**9 - 29*x**8 + 69*x**7 - 124*x**6 + 174*x**5 - 171*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*(-1+x)^7*F(x)-5*x^9+29*x^8-69*x^7+124*x^6-174*x^5+171*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(-1+x \right)^{7} F \! \left(x \right)-5 x^{9}+29 x^{8}-69 x^{7}+124 x^{6}-174 x^{5}+171 x^{4}-110 x^{3}+44 x^{2}-10 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1+7/6*n-7/48*n^3+17/144*n^4-1/48*n^5+1/720*n^6+3/4*2^n+1/4*2^n*n-403/360*n^2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -1+\frac{7 n}{6}-\frac{7 n^{3}}{48}+\frac{17 n^{4}}{144}-\frac{n^{5}}{48}+\frac{n^{6}}{720}+\frac{3 \,2^{n}}{4}+\frac{2^{n} n}{4}-\frac{403 n^{2}}{360} & \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) = 53
a(6) = 134
a(7) = 319
a(8) = 731
a(9) = 1631
a(n+2) = 1/720*n^6-3/80*n^5+47/144*n^4-47/48*n^3-149/180*n^2-4*a(n)+4*a(n+1)+391/60*n-3, 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) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 134
a \! \left(7\right) = 319
a \! \left(8\right) = 731
a \! \left(9\right) = 1631
a \! \left(n +2\right) = \frac{n^{6}}{720}-\frac{3 n^{5}}{80}+\frac{47 n^{4}}{144}-\frac{47 n^{3}}{48}-\frac{149 n^{2}}{180}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{391 n}{60}-3, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21634/
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[14,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[14,x]*F[15,x]*F[18,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[14,x]*F[15,x]
F[14,x] = x
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[14,x]*F[15,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[15,x]*F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[14,x]*F[23,x]
F[23,x] = F[24,x]+F[37,x]
F[24,x] = F[25,x]
F[25,x] = F[18,x]*F[26,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]+F[30,x]
F[28,x] = F[29,x]
F[29,x] = F[14,x]*F[26,x]
F[30,x] = F[31,x]*F[34,x]
F[31,x] = F[32,x]
F[32,x] = F[14,x]*F[33,x]
F[33,x] = F[1,x]+F[32,x]
F[34,x] = F[35,x]
F[35,x] = F[14,x]*F[36,x]
F[36,x] = F[15,x]+F[34,x]
F[37,x] = F[15,x]*F[38,x]
F[38,x] = F[39,x]+F[83,x]
F[39,x] = F[31,x]+F[40,x]
F[40,x] = F[41,x]+F[42,x]+F[45,x]
F[41,x] = 0
F[42,x] = F[14,x]*F[43,x]
F[43,x] = F[34,x]+F[44,x]
F[44,x] = F[42,x]
F[45,x] = F[14,x]*F[46,x]
F[46,x] = F[47,x]+F[60,x]
F[47,x] = F[31,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[14,x]*F[50,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[31,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[14,x]*F[51,x]
F[54,x] = F[55,x]+F[58,x]
F[55,x] = F[56,x]
F[56,x] = F[14,x]*F[57,x]
F[57,x] = F[31,x]+F[55,x]
F[58,x] = F[59,x]
F[59,x] = F[14,x]*F[54,x]
F[60,x] = F[61,x]+F[67,x]
F[61,x] = F[41,x]+F[62,x]+F[65,x]
F[62,x] = F[14,x]*F[63,x]
F[63,x] = F[31,x]+F[64,x]
F[64,x] = F[62,x]
F[65,x] = F[14,x]*F[66,x]
F[66,x] = F[31,x]+F[61,x]
F[67,x] = 2*F[41,x]+F[68,x]+F[70,x]
F[68,x] = F[14,x]*F[69,x]
F[69,x] = F[48,x]+F[67,x]
F[70,x] = F[14,x]*F[71,x]
F[71,x] = F[72,x]+F[77,x]
F[72,x] = F[61,x]+F[73,x]
F[73,x] = 2*F[41,x]+F[74,x]+F[76,x]
F[74,x] = F[14,x]*F[75,x]
F[75,x] = F[52,x]+F[73,x]
F[76,x] = F[14,x]*F[72,x]
F[77,x] = F[78,x]+F[81,x]
F[78,x] = F[79,x]
F[79,x] = F[14,x]*F[80,x]
F[80,x] = F[61,x]+F[78,x]
F[81,x] = F[82,x]
F[82,x] = F[14,x]*F[77,x]
F[83,x] = F[84,x]
F[84,x] = F[14,x]*F[85,x]
F[85,x] = F[16,x]+F[86,x]
F[86,x] = F[84,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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \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_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{37}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right) F_{26}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{83}\! \left(x \right)
F_{39}\! \left(x \right) = F_{31}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = 0
F_{42}\! \left(x \right) = F_{14}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{34}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{14}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{60}\! \left(x \right)
F_{47}\! \left(x \right) = F_{31}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{14}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{31}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{14}\! \left(x \right) F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{14}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{31}\! \left(x \right)+F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{14}\! \left(x \right) F_{54}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{67}\! \left(x \right)
F_{61}\! \left(x \right) = F_{41}\! \left(x \right)+F_{62}\! \left(x \right)+F_{65}\! \left(x \right)
F_{62}\! \left(x \right) = F_{14}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{31}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{14}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{31}\! \left(x \right)+F_{61}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{14}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{48}\! \left(x \right)+F_{67}\! \left(x \right)
F_{70}\! \left(x \right) = F_{14}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{77}\! \left(x \right)
F_{72}\! \left(x \right) = F_{61}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{74}\! \left(x \right)+F_{76}\! \left(x \right)
F_{74}\! \left(x \right) = F_{14}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{52}\! \left(x \right)+F_{73}\! \left(x \right)
F_{76}\! \left(x \right) = F_{14}\! \left(x \right) F_{72}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{81}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{14}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{61}\! \left(x \right)+F_{78}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{14}\! \left(x \right) F_{77}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{14}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{16}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{84}\! \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_14(x)*F_4(x))
Eq(F_4(x), F_21(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_14(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_14(x)*F_15(x)*F_18(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_14(x)*F_15(x))
Eq(F_14(x), x)
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_14(x)*F_15(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_15(x)*F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_14(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_37(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_18(x)*F_26(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x) + F_30(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_14(x)*F_26(x))
Eq(F_30(x), F_31(x)*F_34(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_14(x)*F_33(x))
Eq(F_33(x), F_1(x) + F_32(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_14(x)*F_36(x))
Eq(F_36(x), F_15(x) + F_34(x))
Eq(F_37(x), F_15(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_83(x))
Eq(F_39(x), F_31(x) + F_40(x))
Eq(F_40(x), F_41(x) + F_42(x) + F_45(x))
Eq(F_41(x), 0)
Eq(F_42(x), F_14(x)*F_43(x))
Eq(F_43(x), F_34(x) + F_44(x))
Eq(F_44(x), F_42(x))
Eq(F_45(x), F_14(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_60(x))
Eq(F_47(x), F_31(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_14(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_31(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_14(x)*F_51(x))
Eq(F_54(x), F_55(x) + F_58(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_14(x)*F_57(x))
Eq(F_57(x), F_31(x) + F_55(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_14(x)*F_54(x))
Eq(F_60(x), F_61(x) + F_67(x))
Eq(F_61(x), F_41(x) + F_62(x) + F_65(x))
Eq(F_62(x), F_14(x)*F_63(x))
Eq(F_63(x), F_31(x) + F_64(x))
Eq(F_64(x), F_62(x))
Eq(F_65(x), F_14(x)*F_66(x))
Eq(F_66(x), F_31(x) + F_61(x))
Eq(F_67(x), 2*F_41(x) + F_68(x) + F_70(x))
Eq(F_68(x), F_14(x)*F_69(x))
Eq(F_69(x), F_48(x) + F_67(x))
Eq(F_70(x), F_14(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_77(x))
Eq(F_72(x), F_61(x) + F_73(x))
Eq(F_73(x), 2*F_41(x) + F_74(x) + F_76(x))
Eq(F_74(x), F_14(x)*F_75(x))
Eq(F_75(x), F_52(x) + F_73(x))
Eq(F_76(x), F_14(x)*F_72(x))
Eq(F_77(x), F_78(x) + F_81(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_14(x)*F_80(x))
Eq(F_80(x), F_61(x) + F_78(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_14(x)*F_77(x))
Eq(F_83(x), F_84(x))
Eq(F_84(x), F_14(x)*F_85(x))
Eq(F_85(x), F_16(x) + F_86(x))
Eq(F_86(x), F_84(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": 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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