0132_0213_1032_1302_2310
Counting sequence:
1, 1, 2, 6, 19, 53, 133, 313, 711, 1585, 3497, 7665, 16715, 36281, 78397, 168665, 361359, 771169, 1639761, 3475041, 7341971, 15468777, 32508517, 68160521, 142609879, 297799633, 620761593, 1291850833, 2684360411, 5570042265, 11542731917, 23890763705, 49392132895, 102005483201, 210453408417, 433791708865, 893353210659, 1838246016969, 3779571236021, 7765300888041, 15942918621031, 32710470946097, 67070209315657, 137438953494961, 281474976735339, 576144092981113, 1178676465004253, 2410129488115353, 4925812092468911, 10062730417440545, 20547673299914737, 41939771529926945, 85568392920081075, 174514485560650921, 355784370562316037, 725079540006699465, 1477180677777575159, 3008404551083546769, 6124895493223933081, 12465963768561594641, 25364273101350698491, 51593237331156470617, 104915856919223146797, 213290478352266766201, 433498485732174542399, 880832029519631172993, 1789334175149826594113, 3634008582520781759873, 7378697629483820742211, 14978756187852156012425, 30400234233473341167957, 61685912182484740713385, 125142711796045598277255, 253827198454243430355441, 514737946632791328417257, 1043642992714191592356465, 2115620184325601055870859, 4287908766445637854176569, 8689154328480147193346941, 17604982248138037356810841, 35663311678631560653990351, 72233317721974093188858337, 146280024173370130139617937, 296186825805584147803190241, 599627206528856070654447059, 1213761522893087691405191273, 2456537265456926483003147173, 4971102970255355166392000393, 10058262819193714733555596311, 20348639395753438268654573905, 41161506306238894140396107577, 83251467641941823486966339025, 168359845342811717386281137435, 340433510803479575597259412761, 688294661842671432843913328077, 1391444604156767428986615895865, 2812599769256383984570810513759, 5684620660398466222336778722369, 11488083564568328951063873093601, 23213851616679450914908377752641, 46903072208444487855378018912611
Generating function in Maple syntax:
(x^7+5*x^6-10*x^5+20*x^4-28*x^3+20*x^2-7*x+1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
\frac{x^{7}+5 x^{6}-10 x^{5}+20 x^{4}-28 x^{3}+20 x^{2}-7 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**7 + 5*x**6 - 10*x**5 + 20*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^4*F(x)+x^7+5*x^6-10*x^5+20*x^4-28*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+x^{7}+5 x^{6}-10 x^{5}+20 x^{4}-28 x^{3}+20 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,-1+1/3*n^3-2*n^2+11/3*n+3/8*2^n*n-2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -1+\frac{n^{3}}{3}-2 n^{2}+\frac{11 n}{3}+\frac{3 \,2^{n} n}{8}-2^{-1+n} & \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) = 133
a(7) = 313
a(n+2) = 1/3*n^3-4*n^2-4*a(n)+4*a(n+1)+35/3*n-7, 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) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 133
a \! \left(7\right) = 313
a \! \left(n +2\right) = \frac{n^{3}}{3}-4 n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{35 n}{3}-7, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18350/
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[25,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[23,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[17,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[13,x]
F[25,x] = F[26,x]+F[46,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[1,x]+F[14,x]
F[30,x] = F[18,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]+F[38,x]
F[32,x] = 0
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[10,x]+F[35,x]
F[35,x] = F[32,x]+F[33,x]+F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[18,x]+F[35,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[37,x]+F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[18,x]+F[41,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[40,x]
F[46,x] = F[32,x]+F[47,x]+F[65,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[49,x]+F[56,x]
F[49,x] = F[10,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[10,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[53,x]
F[56,x] = F[35,x]+F[57,x]
F[57,x] = 2*F[32,x]+F[58,x]+F[63,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[54,x]+F[60,x]
F[60,x] = 2*F[32,x]+F[58,x]+F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[35,x]+F[60,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[62,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]+F[70,x]
F[67,x] = F[26,x]+F[68,x]
F[68,x] = F[32,x]+F[47,x]+F[69,x]
F[69,x] = F[12,x]*F[67,x]
F[70,x] = F[71,x]+F[72,x]
F[71,x] = F[32,x]+F[38,x]+F[47,x]
F[72,x] = F[73,x]
F[73,x] = F[12,x]*F[70,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_{25}\! \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_{23}\! \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_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \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_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{46}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{38}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{46}\! \left(x \right) = F_{32}\! \left(x \right)+F_{47}\! \left(x \right)+F_{65}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{56}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \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_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{10}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{35}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{58}\! \left(x \right)+F_{63}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{54}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{58}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{35}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{62}\! \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_{70}\! \left(x \right)
F_{67}\! \left(x \right) = F_{26}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{32}\! \left(x \right)+F_{47}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{72}\! \left(x \right)
F_{71}\! \left(x \right) = F_{32}\! \left(x \right)+F_{38}\! \left(x \right)+F_{47}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \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_25(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_23(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_9(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_17(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_13(x))
Eq(F_25(x), F_26(x) + F_46(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_1(x) + F_14(x))
Eq(F_30(x), F_18(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_38(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_10(x) + F_35(x))
Eq(F_35(x), F_32(x) + F_33(x) + F_36(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_18(x) + F_35(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_37(x) + F_40(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_18(x) + F_41(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_40(x))
Eq(F_46(x), F_32(x) + F_47(x) + F_65(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_56(x))
Eq(F_49(x), F_10(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_10(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_53(x))
Eq(F_56(x), F_35(x) + F_57(x))
Eq(F_57(x), 2*F_32(x) + F_58(x) + F_63(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_54(x) + F_60(x))
Eq(F_60(x), 2*F_32(x) + F_58(x) + F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_35(x) + F_60(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_62(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_70(x))
Eq(F_67(x), F_26(x) + F_68(x))
Eq(F_68(x), F_32(x) + F_47(x) + F_69(x))
Eq(F_69(x), F_12(x)*F_67(x))
Eq(F_70(x), F_71(x) + F_72(x))
Eq(F_71(x), F_32(x) + F_38(x) + F_47(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_12(x)*F_70(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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