0132_0321_1302_3021_3120
Counting sequence:
1, 1, 2, 6, 19, 55, 147, 372, 906, 2143, 4950, 11210, 24973, 54887, 119317, 257096, 550052, 1170111, 2477600, 5226046, 10988159, 23040567, 48198823, 100618492, 209659470, 436138911, 905885658, 1878946258, 3892191297, 8052916679, 16642823401, 34359531536, 70866717064, 146028603263, 300647378948, 618474905862, 1271309875283, 2611339604887, 5360118599771, 10995115609092, 22539987608530, 46179487503583, 94557999012894, 193514045388378, 395824184762421, 809240556656167, 1653665486623869, 3377699718797144, 6896136927484588, 14073748833393215, 28710447622115752, 58546795153194062, 119345390122423655, 243194379874818039, 495395959007248079, 1008806316527142028, 2053641430076728214, 4179340454195205663, 8502796096470455842, 17293822569097207138, 35164105890502845801, 71481133285618001799, 145268109580455648465, 295147905179345157664, 599519182395552121872, 1217485108864821421823, 2471863705877070208780, 5017514388048987563030, 10182602728687661198779, 20660353362554685649751, 41911002535468088192003, 85002596691653599792788, 172366376624742035214746, 349455119732353729637215, 708354972430446764724710, 1435599410792372126415594, 2908977753447701431802845, 5893513370621317205502439, 11938142468694463077603813, 24178516392292583469998184, 48961495694392481549889972, 99131917208399592298529215, 200681686056028442972095536, 406199075390515402670304350, 822069557337947838767295823, 1663481927789729744360765879, 3365649481807127622344934391, 6808670216069591511905893916, 13772082937049855558211131870, 27853650883921056185186224287, 56326271787484802507863522154, 113890483614254985290670121714, 230256847307080731131185000337, 465465454771302983362015678791, 940834429856889008923276326841, 1901475900342344102244993536560, 3842565881941820373286816993176, 7764359926397905084167239065215, 15687176177824338843521630481684, 31691265005705735037417504680230, 64016355311525584775583432491043
Generating function in Maple syntax:
(2*x^8-14*x^7+39*x^6-76*x^5+96*x^4-75*x^3+35*x^2-9*x+1)/(2*x-1)^2/(x-1)^6
Generating function in latex syntax:
\frac{2 x^{8}-14 x^{7}+39 x^{6}-76 x^{5}+96 x^{4}-75 x^{3}+35 x^{2}-9 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{6}}
Generating function in sympy syntax:
(2*x**8 - 14*x**7 + 39*x**6 - 76*x**5 + 96*x**4 - 75*x**3 + 35*x**2 - 9*x + 1)/((x - 1)**6*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^6*F(x)-2*x^8+14*x^7-39*x^6+76*x^5-96*x^4+75*x^3-35*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)-2 x^{8}+14 x^{7}-39 x^{6}+76 x^{5}-96 x^{4}+75 x^{3}-35 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1/120*(n+1)*(n^4-6*n^3+31*n^2+34*n-60*2^n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -\frac{\left(n +1\right) \left(n^{4}-6 n^{3}+31 n^{2}+34 n -60 \,2^{n}\right)}{120} & \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) = 55
a(6) = 147
a(7) = 372
a(8) = 906
a(n+2) = -4*a(n)+4*a(n+1)-1/120*n*(n^4-15*n^3+65*n^2-45*n-246), n >= 9
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) = 55
a \! \left(6\right) = 147
a \! \left(7\right) = 372
a \! \left(8\right) = 906
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{n \left(n^{4}-15 n^{3}+65 n^{2}-45 n -246\right)}{120}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21761/
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[15,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[15,x]*F[7,x]
F[7,x] = F[29,x]+F[8,x]
F[8,x] = F[26,x]*F[9,x]
F[9,x] = F[10,x]+F[24,x]
F[10,x] = F[11,x]+F[19,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[15,x]
F[15,x] = x
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[15,x]*F[16,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[15,x]*F[17,x]*F[21,x]
F[21,x] = F[12,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[15,x]*F[21,x]
F[24,x] = F[25,x]
F[25,x] = F[13,x]*F[15,x]*F[16,x]*F[21,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[15,x]*F[26,x]
F[29,x] = F[16,x]*F[30,x]
F[30,x] = F[13,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]+F[36,x]
F[32,x] = 0
F[33,x] = F[15,x]*F[34,x]
F[34,x] = F[17,x]+F[35,x]
F[35,x] = F[33,x]
F[36,x] = F[15,x]*F[37,x]
F[37,x] = F[38,x]+F[51,x]
F[38,x] = F[13,x]+F[39,x]
F[39,x] = F[32,x]+F[33,x]+F[40,x]
F[40,x] = F[15,x]*F[41,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[13,x]+F[43,x]
F[43,x] = F[32,x]+F[33,x]+F[44,x]
F[44,x] = F[15,x]*F[42,x]
F[45,x] = F[46,x]+F[48,x]
F[46,x] = F[47,x]
F[47,x] = F[13,x]*F[15,x]
F[48,x] = 2*F[32,x]+F[49,x]+F[50,x]
F[49,x] = F[15,x]*F[43,x]
F[50,x] = F[15,x]*F[45,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[53,x]
F[53,x] = F[15,x]*F[54,x]
F[54,x] = F[13,x]+F[52,x]
F[55,x] = 2*F[32,x]+F[56,x]+F[58,x]
F[56,x] = F[15,x]*F[57,x]
F[57,x] = F[39,x]+F[55,x]
F[58,x] = F[15,x]*F[59,x]
F[59,x] = F[60,x]+F[65,x]
F[60,x] = F[52,x]+F[61,x]
F[61,x] = 2*F[32,x]+F[62,x]+F[64,x]
F[62,x] = F[15,x]*F[63,x]
F[63,x] = F[43,x]+F[61,x]
F[64,x] = F[15,x]*F[60,x]
F[65,x] = F[66,x]+F[68,x]
F[66,x] = F[67,x]
F[67,x] = F[15,x]*F[52,x]
F[68,x] = 3*F[32,x]+F[69,x]+F[70,x]
F[69,x] = F[15,x]*F[61,x]
F[70,x] = F[15,x]*F[65,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_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{15}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{29}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{26}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{17}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{21}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{13}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{15}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{17}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{51}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right) F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right) F_{43}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right) F_{45}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{15}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{13}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{15}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{39}\! \left(x \right)+F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{65}\! \left(x \right)
F_{60}\! \left(x \right) = F_{52}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)
F_{62}\! \left(x \right) = F_{15}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{43}\! \left(x \right)+F_{61}\! \left(x \right)
F_{64}\! \left(x \right) = F_{15}\! \left(x \right) F_{60}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{15}\! \left(x \right) F_{52}\! \left(x \right)
F_{68}\! \left(x \right) = 3 F_{32}\! \left(x \right)+F_{69}\! \left(x \right)+F_{70}\! \left(x \right)
F_{69}\! \left(x \right) = F_{15}\! \left(x \right) F_{61}\! \left(x \right)
F_{70}\! \left(x \right) = F_{15}\! \left(x \right) F_{65}\! \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_15(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_15(x)*F_7(x))
Eq(F_7(x), F_29(x) + F_8(x))
Eq(F_8(x), F_26(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_24(x))
Eq(F_10(x), F_11(x) + F_19(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_15(x))
Eq(F_15(x), x)
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_15(x)*F_16(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_15(x)*F_17(x)*F_21(x))
Eq(F_21(x), F_12(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_15(x)*F_21(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_13(x)*F_15(x)*F_16(x)*F_21(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_15(x)*F_26(x))
Eq(F_29(x), F_16(x)*F_30(x))
Eq(F_30(x), F_13(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_36(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_15(x)*F_34(x))
Eq(F_34(x), F_17(x) + F_35(x))
Eq(F_35(x), F_33(x))
Eq(F_36(x), F_15(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_51(x))
Eq(F_38(x), F_13(x) + F_39(x))
Eq(F_39(x), F_32(x) + F_33(x) + F_40(x))
Eq(F_40(x), F_15(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_13(x) + F_43(x))
Eq(F_43(x), F_32(x) + F_33(x) + F_44(x))
Eq(F_44(x), F_15(x)*F_42(x))
Eq(F_45(x), F_46(x) + F_48(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_13(x)*F_15(x))
Eq(F_48(x), 2*F_32(x) + F_49(x) + F_50(x))
Eq(F_49(x), F_15(x)*F_43(x))
Eq(F_50(x), F_15(x)*F_45(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_15(x)*F_54(x))
Eq(F_54(x), F_13(x) + F_52(x))
Eq(F_55(x), 2*F_32(x) + F_56(x) + F_58(x))
Eq(F_56(x), F_15(x)*F_57(x))
Eq(F_57(x), F_39(x) + F_55(x))
Eq(F_58(x), F_15(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_65(x))
Eq(F_60(x), F_52(x) + F_61(x))
Eq(F_61(x), 2*F_32(x) + F_62(x) + F_64(x))
Eq(F_62(x), F_15(x)*F_63(x))
Eq(F_63(x), F_43(x) + F_61(x))
Eq(F_64(x), F_15(x)*F_60(x))
Eq(F_65(x), F_66(x) + F_68(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_15(x)*F_52(x))
Eq(F_68(x), 3*F_32(x) + F_69(x) + F_70(x))
Eq(F_69(x), F_15(x)*F_61(x))
Eq(F_70(x), F_15(x)*F_65(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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