0132_0213_0231_1320_2103
Counting sequence:
1, 1, 2, 6, 19, 53, 134, 328, 801, 1966, 4843, 11945, 29463, 72656, 179144, 441685, 1088988, 2684959, 6619951, 16321973, 40243014, 99222044, 244639025, 603174922, 1487170695, 3666725245, 9040572271, 22290174944, 54958014156, 135502898717, 334092048920, 823727744563, 2030959429803, 5007475143157, 12346286657302, 30440649202832, 75053588954289, 185049969774902, 456253881936659, 1124926445734993, 2773586282581383, 6838474547460544, 16860746114140240, 41571370567002389, 102497175339689988, 252714086866168727, 623084582467232135, 1536259421557154869, 3787757034490964982, 9338984777579781988, 23025932202535767649, 56772076025717055602, 139975597422928730127, 345119806170736945445, 850917465645455297951, 2097997623996922654960, 5172762586272624966420, 12753814622042252484733, 31445438428796653071712, 77530968363793224762443, 191158125177316391145411, 471313974176689399984565, 1162058175911555318498534, 2865137208294723447507544, 7064199876151185397491201, 17417287990865790307922686, 42943564207591404586757371, 105880416504490222999889657, 261055708952601239989539831, 643651445910728058718983152, 1586968488393376703927902424, 3912783850877670242085313429, 9647247299275981624858691052, 23785975407895705400791904335, 58646016687835924205130102367, 144595931609693114691136960949, 356511569223279953941273044966, 879004668908162014924797682220, 2167248624345327634066236386897, 5343505860510340323464776271386, 13174794327039209168302316197719, 32483393878641117958084092594637, 80090121460899446961219118204111, 197467899431509825559362958436416, 486871172057465421305608520509020, 1200415555455010898309276644466333, 2959710059827245963387152850629480, 7297375978206325625919053741713315, 17992200280054105381668280647104859, 44361051408666531287239312431741301, 109375332169014575118373007716580854, 269672672472887960182300310648931488, 664897183272478558999054141308045585, 1639351367232518160439834035137267462, 4041937569986343727310966365955636387, 9965684993600219784041857712060344993, 24571106226166714980932101123875000423, 60581812646624792185452015033065185888, 149368774436466253189273127649066776416, 368279353191241196772628738802566177493, 908018978522507897700505362064126385556
Generating function in Maple syntax:
(x^6-x^5-2*x^4-3*x^2+3*x-1)/(3*x^3-5*x^2+4*x-1)
Generating function in latex syntax:
\frac{x^{6}-x^{5}-2 x^{4}-3 x^{2}+3 x -1}{3 x^{3}-5 x^{2}+4 x -1}
Generating function in sympy syntax:
(x**6 - x**5 - 2*x**4 - 3*x**2 + 3*x - 1)/(3*x**3 - 5*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(-3*x^3+5*x^2-4*x+1)*F(x)+x^6-x^5-2*x^4-3*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(-3 x^{3}+5 x^{2}-4 x +1\right) F \! \left(x \right)+x^{6}-x^{5}-2 x^{4}-3 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
-1309/837*Sum(_alpha^(1-n),_alpha = RootOf(3*_Z^3-5*_Z^2+4*_Z-1))+1853/837*Sum(_alpha^(-n),_alpha = RootOf(3*_Z^3-5*_Z^2+4*_Z-1))-338/837*Sum(_alpha^(-n-1),_alpha = RootOf(3*_Z^3-5*_Z^2+4*_Z-1))+1/3*piecewise(n = 0,-115/27,n = 1,-20/9,n = 2,2/3,n = 3,1,0)
Explicit closed form in latex syntax:
-\frac{1309 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{837}+\frac{1853 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{837}-\frac{338 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{837}+\frac{\left(\left\{\begin{array}{cc}-\frac{115}{27} & n =0 \\ -\frac{20}{9} & n =1 \\ \frac{2}{3} & n =2 \\ 1 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)}{3}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(6) = 134
a(n+3) = 3*a(n)-5*a(n+1)+4*a(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) = 53
a \! \left(6\right) = 134
a \! \left(n +3\right) = 3 a \! \left(n \right)-5 a \! \left(n +1\right)+4 a \! \left(n +2\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18179/
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[14,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[7,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[49,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[30,x]
F[19,x] = F[20,x]+F[7,x]
F[20,x] = F[16,x]+F[21,x]+F[26,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]
F[23,x] = F[11,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[23,x]*F[4,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[13,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[21,x]
F[30,x] = F[31,x]+F[37,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]
F[34,x] = F[2,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[34,x]*F[4,x]
F[37,x] = 2*F[16,x]+F[38,x]+F[43,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[35,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[29,x]+F[7,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[62,x]
F[51,x] = F[2,x]+F[52,x]
F[52,x] = F[16,x]+F[53,x]+F[61,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[58,x]
F[55,x] = F[11,x]+F[56,x]
F[56,x] = F[16,x]+F[21,x]+F[57,x]
F[57,x] = F[13,x]*F[4,x]
F[58,x] = F[35,x]+F[59,x]
F[59,x] = 2*F[16,x]+F[38,x]+F[60,x]
F[60,x] = F[4,x]*F[45,x]
F[61,x] = F[4,x]*F[51,x]
F[62,x] = F[46,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_{14}\! \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_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{49}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{30}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{26}\! \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_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{37}\! \left(x \right)
F_{31}\! \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_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{35}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \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_{62}\! \left(x \right)
F_{51}\! \left(x \right) = F_{2}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{16}\! \left(x \right)+F_{53}\! \left(x \right)+F_{61}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{55}\! \left(x \right) = F_{11}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{58}\! \left(x \right) = F_{35}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{62}\! \left(x \right) = F_{46}\! \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_14(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_7(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_49(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_30(x))
Eq(F_19(x), F_20(x) + F_7(x))
Eq(F_20(x), F_16(x) + F_21(x) + F_26(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_11(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_23(x)*F_4(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_13(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_21(x))
Eq(F_30(x), F_31(x) + F_37(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_2(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_34(x)*F_4(x))
Eq(F_37(x), 2*F_16(x) + F_38(x) + F_43(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_35(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_29(x) + F_7(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_62(x))
Eq(F_51(x), F_2(x) + F_52(x))
Eq(F_52(x), F_16(x) + F_53(x) + F_61(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_58(x))
Eq(F_55(x), F_11(x) + F_56(x))
Eq(F_56(x), F_16(x) + F_21(x) + F_57(x))
Eq(F_57(x), F_13(x)*F_4(x))
Eq(F_58(x), F_35(x) + F_59(x))
Eq(F_59(x), 2*F_16(x) + F_38(x) + F_60(x))
Eq(F_60(x), F_4(x)*F_45(x))
Eq(F_61(x), F_4(x)*F_51(x))
Eq(F_62(x), F_46(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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