0213_1230_1302
Counting sequence:
1, 1, 2, 6, 21, 74, 255, 857, 2815, 9063, 28677, 89389, 275034, 836689, 2520128, 7524372, 22291317, 65581700, 191743857, 557466281, 1612517029, 4642784877, 13311163941, 38016320047, 108187223460, 306869427745, 867780176654, 2447031902418, 6882225083349, 19308762485774, 54048777806931, 150968449529465, 420835777861723, 1170893635457955, 3251982915176517, 9016723928520193, 24960835429100766, 68994776676731761, 190438223486935916, 524933540135529792, 1445093576633102133, 3973343537175575528, 10912139291739364005, 29935101489516403913, 82033350165107382049, 224573977463040824841, 614195678748150295653, 1678225407674946690979, 4581490561627535276808, 12496604094171405355585, 34058125428646942162010, 92748506206941317353566, 252385093382277143942037, 686283365343197882294546, 1864824449359332751363431, 5063819624695941324321689, 13741520210892904677005815, 37266409752265492684044447, 101003396268846174696507525, 273589231279844059362821653, 740653578151275463009526562, 2003975666769610667791325329, 5419243343475326079747273944, 14647418323269038818114274604, 39570061950505533490502983413, 106846981281861625533709097612, 288372624687210810058128916377, 777943636228234691091027474665, 2097733806172341085660480868317, 5654136980403947487414444459621, 15233556297705780710580821850789, 41026116099044945559774328852567, 110445625994605394494140997138476, 297214735584064726087596058506337, 799521874961733015137624274512870, 2149972363618497817961664673277610, 5779390833752939475971234680565589, 15530373418829968248343877170044758, 41719222865123918450398259704960059, 112033701970615092893274665082530489, 300761940014183416770378984614004691, 807164772515033204418555728123800155, 2165558624869474738100167812108968133, 5808303322261362776018893280595600745, 15574111811889426454477717115280043302, 41747925340996357549723671517285440433, 111878585480582622514023316029200851076, 299738303813846422743913125094835285976, 802829627170775215623822221510187000501, 2149774819502655020250625877299024299632, 5755107883863326853089392725317459938893, 15403069821944030336116850596745289123593, 41215136092936722057114566254158604136665, 110256468707602776729899961573880579501953, 294884184800258008354870972308362335508709, 788498827633337642441043775744180262112331, 2107923434259954768627288358292010389411088, 5633973489137477186664391104653255502269569, 15055073156206593548873695020266996179268018, 40221744638927580953392123682821740645530550, 107436216311438033229011734507411226375034645
Generating function in Maple syntax:
(x^6-4*x^5+12*x^4-27*x^3+23*x^2-8*x+1)/(x^2-3*x+1)^3
Generating function in latex syntax:
\frac{x^{6}-4 x^{5}+12 x^{4}-27 x^{3}+23 x^{2}-8 x +1}{\left(x^{2}-3 x +1\right)^{3}}
Generating function in sympy syntax:
(x**6 - 4*x**5 + 12*x**4 - 27*x**3 + 23*x**2 - 8*x + 1)/(x**2 - 3*x + 1)**3
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)^3*F(x)-x^6+4*x^5-12*x^4+27*x^3-23*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right)^{3} F \! \left(x \right)-x^{6}+4 x^{5}-12 x^{4}+27 x^{3}-23 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/500*((15*n^2-185*n+36)*5^(1/2)-25*n^2+415*n)*(3/2-1/2*5^(1/2))^(-n)-3/100*((n^2-37/3*n+12/5)*5^(1/2)+5/3*n^2-83/3*n)*(3/2+1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(15 n^{2}-185 n +36\right) \sqrt{5}-25 n^{2}+415 n \right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{500}-\\\frac{3 \left(\left(n^{2}-\frac{37}{3} n +\frac{12}{5}\right) \sqrt{5}+\frac{5 n^{2}}{3}-\frac{83 n}{3}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{100} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 74
a(6) = 255
a(n+6) = -a(n)+9*a(n+1)-30*a(n+2)+45*a(n+3)-30*a(n+4)+9*a(n+5), 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) = 21
a \! \left(5\right) = 74
a \! \left(6\right) = 255
a \! \left(n +6\right) = -a \! \left(n \right)+9 a \! \left(n +1\right)-30 a \! \left(n +2\right)+45 a \! \left(n +3\right)-30 a \! \left(n +4\right)+9 a \! \left(n +5\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/327/
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[40,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[26,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[25,x]
F[20,x] = 0
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[12,x]
F[24,x] = F[19,x]
F[25,x] = F[12,x]*F[14,x]
F[26,x] = F[20,x]+F[27,x]+F[39,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[10,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[10,x]*F[12,x]
F[32,x] = F[26,x]+F[33,x]
F[33,x] = 2*F[20,x]+F[34,x]+F[38,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[30,x]
F[37,x] = F[33,x]
F[38,x] = F[12,x]*F[26,x]
F[39,x] = F[12,x]*F[13,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[56,x]
F[44,x] = F[45,x]
F[45,x] = F[46,x]*F[51,x]
F[46,x] = F[47,x]+F[5,x]
F[47,x] = F[48,x]*F[6,x]
F[48,x] = F[1,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[48,x]
F[51,x] = F[52,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[9,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[48,x]*F[53,x]*F[9,x]
F[56,x] = F[48,x]*F[57,x]
F[57,x] = F[40,x]+F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[51,x]*F[6,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_{40}\! \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_{26}\! \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_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{20}\! \left(x \right) = 0
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_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)+F_{39}\! \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_{32}\! \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_{31}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{32}\! \left(x \right) = F_{26}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{30}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \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_{44}\! \left(x \right)+F_{56}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right) F_{51}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{5}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right) F_{6}\! \left(x \right)
F_{48}\! \left(x \right) = F_{1}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{9}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right) F_{53}\! \left(x \right) F_{9}\! \left(x \right)
F_{56}\! \left(x \right) = F_{48}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{40}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{51}\! \left(x \right) F_{6}\! \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_40(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_26(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_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_25(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_12(x))
Eq(F_24(x), F_19(x))
Eq(F_25(x), F_12(x)*F_14(x))
Eq(F_26(x), F_20(x) + F_27(x) + F_39(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_10(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_10(x)*F_12(x))
Eq(F_32(x), F_26(x) + F_33(x))
Eq(F_33(x), 2*F_20(x) + F_34(x) + F_38(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_30(x))
Eq(F_37(x), F_33(x))
Eq(F_38(x), F_12(x)*F_26(x))
Eq(F_39(x), F_12(x)*F_13(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_56(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_46(x)*F_51(x))
Eq(F_46(x), F_47(x) + F_5(x))
Eq(F_47(x), F_48(x)*F_6(x))
Eq(F_48(x), F_1(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_48(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_9(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_48(x)*F_53(x)*F_9(x))
Eq(F_56(x), F_48(x)*F_57(x))
Eq(F_57(x), F_40(x) + F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_51(x)*F_6(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": 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