0231_0321_1032_1203_1230
Counting sequence:
1, 1, 2, 6, 19, 55, 151, 401, 1042, 2666, 6742, 16896, 42041, 104013, 256165, 628575, 1537824, 3753296, 9142628, 22235054, 54005831, 131033011, 317644563, 769467693, 1862871214, 4507792822, 10903622450, 26365369116, 63735023701, 154036742809, 372211162177, 899264373179, 2172350520892, 5247186640028, 12673166251456, 30606404044362, 73911744143459, 178481431938303, 430977687234607, 1040642964837129, 2512675933768650, 6066819466094594, 14647964133398798, 35366046267774776, 85386653738714225, 206152547884735717, 497718137787251421, 1201641600017370903, 2901106890938258776, 7004066488126420456, 16909662079656164636, 40824235072368880614, 98559821074254188671, 237947254920597784651, 574461086314890812491, 1386882938349261519845, 3348253984611178073830, 8083444950767146112078, 19515251972536527188458, 47114165068622314271316, 113744014455345383297165, 274603058670441535999281, 662951861178485365564537, 1600510239791926087667571, 3863979258291365181979764, 9328482591432711733788980, 22520972111272899213883256, 54370482154210731290208386, 131262047100158804051607707, 316894797715457223908041111, 765052085252931020896926567, 1846999853665034803760369601, 4459053563470431704534618626, 10765110522380760365063514778, 25989281691781676739129466310, 62743688073043562452258086000, 151476686172067698861516917929, 365697117085576754611034483581, 882871033680016796955071011093, 2131439411119201526264146760399, 5145750309265602204969305043856, 12422940936344770647174637874816, 29991633995343872921262342849748, 72406212553809975333586847689758, 174804066356518741276211086463159, 422014359773957293261559117086947, 1018832814918652998550429513581955, 2459680047639702631864618530143901, 5938193026254936945284067345659166, 14336066332263333888441554765044454, 34610326155009119454184779962916002, 83556719570736602260846320865215884, 201723767153392382903947834042030277, 487004257591341485924882813646645193, 1175732289763715590465995110730062001, 2838468851974053138281436333896255787, 6852670023422382809877994376101550764, 16543808958239940643735678281257315788, 39940288058744507868745857328932103472, 96424385313413443924020405719753369274, 232789059160940370802372694329702539475
Generating function in Maple syntax:
-(3*x^6-10*x^3+13*x^2-6*x+1)/(x^2+2*x-1)/(2*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{3 x^{6}-10 x^{3}+13 x^{2}-6 x +1}{\left(x^{2}+2 x -1\right) \left(2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-3*x**6 + 10*x**3 - 13*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+2*x-1)*(2*x-1)*(x-1)^3*F(x)+3*x^6-10*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+2 x -1\right) \left(2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+3 x^{6}-10 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/8*(-2*2^(1/2)+7)*(-1-2^(1/2))^(-n)+1/8*(2*2^(1/2)+7)*(2^(1/2)-1)^(-n)+1/4*n^2-5/4*n-3*2^(n-1)+9/4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-2 \sqrt{2}+7\right) \left(-1-\sqrt{2}\right)^{-n}}{8}+\frac{\left(2 \sqrt{2}+7\right) \left(\sqrt{2}-1\right)^{-n}}{8}+\frac{n^{2}}{4}-\frac{5 n}{4}-\\3 \,2^{n -1}+\frac{9}{4} & \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) = 151
a(n+3) = -2*a(n)-3*a(n+1)+4*a(n+2)+1/2*(n-4)*(n-3), 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) = 55
a \! \left(6\right) = 151
a \! \left(n +3\right) = -2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)+\frac{\left(n -4\right) \left(n -3\right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20755/
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[10,x]*F[4,x]
F[4,x] = F[19,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = x
F[11,x] = F[12,x]+F[13,x]+F[17,x]
F[12,x] = 0
F[13,x] = F[10,x]*F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[10,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[10,x]*F[18,x]
F[18,x] = F[9,x]
F[19,x] = F[20,x]+F[41,x]
F[20,x] = F[21,x]
F[21,x] = F[10,x]*F[22,x]
F[22,x] = F[23,x]+F[27,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[10,x]*F[26,x]
F[26,x] = F[1,x]+F[10,x]
F[27,x] = F[20,x]+F[28,x]
F[28,x] = F[12,x]+F[29,x]+F[39,x]
F[29,x] = F[10,x]*F[30,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[10,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = x^2
F[34,x] = F[35,x]+F[37,x]
F[35,x] = F[12,x]+F[29,x]+F[36,x]
F[36,x] = F[10,x]*F[20,x]
F[37,x] = F[38,x]
F[38,x] = F[10,x]*F[35,x]
F[39,x] = F[10,x]*F[40,x]
F[40,x] = F[20,x]+F[35,x]
F[41,x] = F[12,x]+F[42,x]+F[57,x]
F[42,x] = F[10,x]*F[43,x]
F[43,x] = F[44,x]+F[49,x]
F[44,x] = F[14,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[10,x]*F[47,x]
F[47,x] = F[10,x]+F[48,x]
F[48,x] = F[46,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[12,x]+F[42,x]+F[51,x]
F[51,x] = F[10,x]*F[52,x]
F[52,x] = F[20,x]+F[50,x]
F[53,x] = F[54,x]
F[54,x] = F[10,x]*F[55,x]
F[55,x] = F[35,x]+F[56,x]
F[56,x] = F[54,x]
F[57,x] = F[10,x]*F[58,x]
F[58,x] = F[19,x]+F[59,x]
F[59,x] = F[35,x]+F[60,x]
F[60,x] = F[12,x]+F[61,x]+F[62,x]+F[63,x]
F[61,x] = 0
F[62,x] = F[10,x]*F[50,x]
F[63,x] = F[10,x]*F[64,x]
F[64,x] = F[59,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{19}\! \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_{10}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{12}\! \left(x \right) = 0
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{9}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{41}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{27}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right)+F_{29}\! \left(x \right)+F_{39}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{10}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = x^{2}
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right)+F_{29}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{20}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right) F_{35}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{20}\! \left(x \right)+F_{35}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right)+F_{42}\! \left(x \right)+F_{57}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{44}\! \left(x \right) = F_{14}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right)+F_{42}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{10}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{20}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{10}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{35}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{10}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{19}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{35}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right)+F_{61}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = 0
F_{62}\! \left(x \right) = F_{10}\! \left(x \right) F_{50}\! \left(x \right)
F_{63}\! \left(x \right) = F_{10}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{59}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_19(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_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_13(x) + F_17(x))
Eq(F_12(x), 0)
Eq(F_13(x), F_10(x)*F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_10(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_10(x)*F_18(x))
Eq(F_18(x), F_9(x))
Eq(F_19(x), F_20(x) + F_41(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_10(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_27(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_10(x)*F_26(x))
Eq(F_26(x), F_1(x) + F_10(x))
Eq(F_27(x), F_20(x) + F_28(x))
Eq(F_28(x), F_12(x) + F_29(x) + F_39(x))
Eq(F_29(x), F_10(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_10(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), x**2)
Eq(F_34(x), F_35(x) + F_37(x))
Eq(F_35(x), F_12(x) + F_29(x) + F_36(x))
Eq(F_36(x), F_10(x)*F_20(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_10(x)*F_35(x))
Eq(F_39(x), F_10(x)*F_40(x))
Eq(F_40(x), F_20(x) + F_35(x))
Eq(F_41(x), F_12(x) + F_42(x) + F_57(x))
Eq(F_42(x), F_10(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_49(x))
Eq(F_44(x), F_14(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_10(x)*F_47(x))
Eq(F_47(x), F_10(x) + F_48(x))
Eq(F_48(x), F_46(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_12(x) + F_42(x) + F_51(x))
Eq(F_51(x), F_10(x)*F_52(x))
Eq(F_52(x), F_20(x) + F_50(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_10(x)*F_55(x))
Eq(F_55(x), F_35(x) + F_56(x))
Eq(F_56(x), F_54(x))
Eq(F_57(x), F_10(x)*F_58(x))
Eq(F_58(x), F_19(x) + F_59(x))
Eq(F_59(x), F_35(x) + F_60(x))
Eq(F_60(x), F_12(x) + F_61(x) + F_62(x) + F_63(x))
Eq(F_61(x), 0)
Eq(F_62(x), F_10(x)*F_50(x))
Eq(F_63(x), F_10(x)*F_64(x))
Eq(F_64(x), F_59(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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