0123_0132_1023_1302_1320

Counting sequence:
1, 1, 2, 6, 19, 54, 146, 382, 979, 2474, 6190, 15374, 37971, 93374, 228826, 559230, 1363683, 3319378, 8067990, 19586446, 47503043, 115116838, 278785314, 674784638, 1632548915, 3948271098, 9545868350, 23073562254, 55760101747, 134727983502, 325484504234, 786233862910, 1899025971907, 4586433290402, 11076187520038, 26747398265102, 64588163919459, 155958085842422, 376573055081074, 909241634958078, 2195331202904211, 5300453796580426, 12797338307692878, 30897329435221774, 74596395224647571, 180098915977539166, 434811819365770362, 1049757739081168766, 2534397666272285603, 6118693809114095346, 14772066759477186998, 35663390278021890702, 86099973215427811075, 207865588508691198150, 501835653832437577922, 1211545903372821095038, 2924945474976589250035, 7061472882123018559130, 17047963296816664296286, 41157543590944423007630, 99363338709081662023347, 239884797469860050477870, 579134086570306369826122, 1398155276453482003824126, 3375449251163288804862339, 8149063002152096468324674, 19673593702211555451063366, 47496287300063354789554702, 114666242089314559868379299, 276828919052645064202726294, 668324375342509867626657810, 1613478260033475158161693694, 3895282076001080901361348691, 9404044773218878395706997994, 22703376344805320562420558446, 54810806907562485259838542350, 132325009049396222560678498003, 319460862785286793338357247998, 771246810177833535151716413210, 1861954634256681315470436912766, 4495156380922651069749883915363, 10852268000564893262284792096658, 26199693590978257208948642814934, 63251657600373046909440427138958, 152703013627427629486346195917635, 368657694526634862799166216623718, 890018422023510468918745424463970, 2148694577259282028304790656149374, 5187407653913326980864593917958067, 12523510039828440900706512854456122, 30234428043055218603622688351651454, 72992366744908897750642027007321230, 176219162770813053390287017265418227, 425430694762415083101976611336406222, 1027080557247403376735761339534727658, 2479591819160742150856541489598855422, 5986264215375928307014928717118426179, 14452120289526680022018567720607683042, 34890504873657450865316401751877742694, 84233130195297906781180046411451069198, 203356765581165914484733844948955682531

Generating function in Maple syntax:
(2*x^6-3*x^5-2*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^{6}-3 x^{5}-2 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**6 - 3*x**5 - 2*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^6-3*x^5-2*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^{6}-3 x^{5}-2 x^{4}-2 x^{3}+8 x^{2}-5 x +1 = 0

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/8*(2^(1/2)+10)*(-1-2^(1/2))^(-n)+1/8*(-2^(1/2)+10)*(2^(1/2)-1)^(-n)-1/2*n-2^n+1/2)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(\sqrt{2}+10\right) \left(-1-\sqrt{2}\right)^{-n}}{8}+\frac{\left(-\sqrt{2}+10\right) \left(\sqrt{2}-1\right)^{-n}}{8}-\frac{n}{2}-2^{n}+\frac{1}{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) = 54
a(6) = 146
a(n+3) = -2*a(n)-3*a(n+1)+4*a(n+2)-n+2, 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) = 54
a \! \left(6\right) = 146
a \! \left(n +3\right) = -2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)-n +2, \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16311/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[12,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[57,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[30,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[1,x]+F[24,x]
F[27,x] = F[13,x]+F[14,x]+F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[7,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[13,x]+F[21,x]+F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[2,x]
F[34,x] = F[13,x]+F[35,x]+F[41,x]+F[56,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[24,x]
F[40,x] = F[35,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[13,x]+F[44,x]+F[55,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[12,x]+F[4,x]
F[47,x] = F[43,x]+F[48,x]
F[48,x] = F[13,x]+F[41,x]+F[49,x]+F[54,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[53,x]
F[51,x] = F[52,x]
F[52,x] = F[24,x]*F[4,x]
F[53,x] = F[49,x]
F[54,x] = 0
F[55,x] = F[2,x]*F[4,x]
F[56,x] = 0
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[2,x]+F[43,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{57}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \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_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{7}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right)
F_{34}\! \left(x \right) = F_{13}\! \left(x \right)+F_{35}\! \left(x \right)+F_{41}\! \left(x \right)+F_{56}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{24}\! \left(x \right)
F_{40}\! \left(x \right) = F_{35}\! \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_{43}\! \left(x \right) = F_{13}\! \left(x \right)+F_{44}\! \left(x \right)+F_{55}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{13}\! \left(x \right)+F_{41}\! \left(x \right)+F_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = 0
F_{55}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = 0
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{2}\! \left(x \right)+F_{43}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_57(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_30(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_1(x) + F_24(x))
Eq(F_27(x), F_13(x) + F_14(x) + F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_7(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_13(x) + F_21(x) + F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_2(x))
Eq(F_34(x), F_13(x) + F_35(x) + F_41(x) + F_56(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_24(x))
Eq(F_40(x), F_35(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_13(x) + F_44(x) + F_55(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_12(x) + F_4(x))
Eq(F_47(x), F_43(x) + F_48(x))
Eq(F_48(x), F_13(x) + F_41(x) + F_49(x) + F_54(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_53(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_24(x)*F_4(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), 0)
Eq(F_55(x), F_2(x)*F_4(x))
Eq(F_56(x), 0)
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_2(x) + F_43(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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