0213_1230_1302_3102
Counting sequence:
1, 1, 2, 6, 20, 63, 185, 518, 1409, 3769, 9987, 26324, 69182, 181519, 475828, 1246664, 3265254, 8550813, 22389903, 58623228, 153486715, 401848051, 1052075357, 2754406906, 7211191980, 18879244333, 49426662710, 129400940538, 338776477064, 886929005259, 2322011371157, 6079106454914, 15915310172357, 41666827587229, 109085178292743, 285588716519024, 747680986195274, 1957454266225243, 5124681851569288, 13416591351729308, 35125092305953530, 91958685731712201, 240750965157098187, 630294210173077656, 1650131666063544415, 4320100789152459703, 11310170703230147585, 29610411323509199158, 77521063272104977944, 202952778500584477849, 531337272242234725802, 1391059038246484711854, 3641839842530170691132, 9534460489397343654039, 24961541625748127843633, 65350164387986623740734, 171088951538437594813769, 447916690227691595998273, 1172661119145228479912523, 3070066667208950565766988, 8037538882483171226146070, 21042549980243067843454951, 55090111058250085043758492, 144227783194513744758142256, 377593238525301759440527950, 988551932381408701243621173, 2588062558618952122180372935, 6775635743475492610867712628, 17738844671807598433883015299, 46440898271947420359811796539, 121583850144034853038043085797, 318310652160157446815838633394, 833348106336437985863484696132, 2181733666849157317290148506949, 5711852894211035270976505755998, 14953825015783950607124446741794, 39149622153140819966851457378864, 102495041443638514821369626282403, 268335502177774733441651745262733, 702511465089685699975919634185018, 1839198893091282389902835505763085, 4815085214184161507621649256251301, 12606056749461202194267902984605647, 33003085034199445174376912792324456, 86403198353137133489363479208738210, 226206510025211955553409021745016243, 592216331722498733591059726753803760, 1550442485142284245899661796155010948, 4059111123704354005208013440077334754, 10626890885970777771504357940081711329, 27821561534207979312185127574538619267, 72837793716653160169711071393909680784, 190691819615751501204488200411936773607, 499237665130601343455953690257022520959, 1307021175776052529183113144578999016713, 3421825862197556244125326178114964633454, 8958456410816616203244546098620753211184, 23453543370252292365691933261237143427633, 61402173699940260893966555537435383822418, 160752977729568490316426656346903563213398, 420856759488765210055667638351454567737700
Generating function in Maple syntax:
(x^7-3*x^6+3*x^5-10*x^3+13*x^2-6*x+1)/(x^2-3*x+1)/(x^2+x-1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{7}-3 x^{6}+3 x^{5}-10 x^{3}+13 x^{2}-6 x +1}{\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**7 - 3*x**6 + 3*x**5 - 10*x**3 + 13*x**2 - 6*x + 1)/((x - 1)**3*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^2+x-1)*(x-1)^3*F(x)-x^7+3*x^6-3*x^5+10*x^3-13*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)-x^{7}+3 x^{6}-3 x^{5}+10 x^{3}-13 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/10*(2*5^(1/2)-5)*(-1/2*5^(1/2)-1/2)^(-n)+1/10*(-2*5^(1/2)-5)*(1/2*5^(1/2)-1/2)^(-n)-1/2*n^2+3/10*(3/2-1/2*5^(1/2))^(-n)*5^(1/2)-3/10*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)+1/2*n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(2 \sqrt{5}-5\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}+\frac{\left(-2 \sqrt{5}-5\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}-\frac{n^{2}}{2}+\\\frac{3 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{10}-\frac{3 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{10}+\frac{n}{2}+1 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(6) = 185
a(7) = 518
a(n+4) = -1/2*n^2-2*a(n+1)-3*a(n+2)+4*a(n+3)+a(n)+1/2*n+4, n >= 8
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) = 20
a \! \left(5\right) = 63
a \! \left(6\right) = 185
a \! \left(7\right) = 518
a \! \left(n +4\right) = -\frac{n^{2}}{2}-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right)+a \! \left(n \right)+\frac{n}{2}+4, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15759/
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[16,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[6,x]
F[10,x] = F[11,x]+F[7,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[7,x]
F[15,x] = F[11,x]
F[16,x] = F[17,x]+F[7,x]
F[17,x] = F[18,x]+F[19,x]+F[50,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[31,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[12,x]+F[18,x]+F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[21,x]+F[25,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[11,x]
F[30,x] = F[27,x]
F[31,x] = F[17,x]+F[32,x]
F[32,x] = 2*F[18,x]+F[33,x]+F[46,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[42,x]
F[38,x] = F[39,x]+F[7,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[38,x]
F[42,x] = F[35,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[35,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[11,x]
F[49,x] = F[27,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[35,x]+F[7,x]
F[53,x] = F[11,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_{16}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{7}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{7}\! \left(x \right)
F_{15}\! \left(x \right) = F_{11}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{50}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{31}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \left(x \right)
F_{31}\! \left(x \right) = F_{17}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{46}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{7}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{38}\! \left(x \right)
F_{42}\! \left(x \right) = F_{35}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{35}\! \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_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{11}\! \left(x \right)
F_{49}\! \left(x \right) = F_{27}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{35}\! \left(x \right)+F_{7}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \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_16(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_6(x))
Eq(F_10(x), F_11(x) + F_7(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), F_7(x))
Eq(F_15(x), F_11(x))
Eq(F_16(x), F_17(x) + F_7(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_50(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_31(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_12(x) + F_18(x) + F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_21(x) + F_25(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_11(x))
Eq(F_30(x), F_27(x))
Eq(F_31(x), F_17(x) + F_32(x))
Eq(F_32(x), 2*F_18(x) + F_33(x) + F_46(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_42(x))
Eq(F_38(x), F_39(x) + F_7(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_38(x))
Eq(F_42(x), F_35(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_35(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_11(x))
Eq(F_49(x), F_27(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_35(x) + F_7(x))
Eq(F_53(x), F_11(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, 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]]}, {"patt": [3, 1, 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]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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