0213_1230_1302_3021
Counting sequence:
1, 1, 2, 6, 20, 62, 179, 495, 1336, 3558, 9406, 24764, 65047, 170629, 447238, 1171718, 3068928, 8036690, 21043755, 55098787, 144259404, 377690406, 988829562, 2588826976, 6777697775, 17744341417, 46455447914, 121622198790, 318411466316, 833612714438, 2182427509091, 5713671159159, 14958588146752, 39162096805734, 102527707973398, 268421036341988, 702735415982983, 1839785235764845, 4816620330399790, 12610075818680582, 33013607227976184, 86430746030828186, 226278631132422747, 592405147799934571, 1550936812968789780, 4060405292241338022, 10630279065591536274, 27830431907504485960, 72861016661729448671, 190752618085462602193, 499396837607244627026, 1307437894756636290054, 3422916846695615523332, 8961312645383526571214, 23461021089541231761683, 61421750623379752576383, 160804230780823877401288, 420990941719457314923750, 1102168594378139354099950, 2885514841415917469402252, 7554375929871161062862839, 19777612948200070449968341, 51778462914733103026580182, 135557775796005796100092166, 354894864473294895483554160, 929126817623896058030748002, 2432485588398421056498725259, 6368329947571412057035640755, 16672504254315887838068445276, 43649182815376369126200156198, 114275044191813409933022732586, 299175949760064168734389211824, 783252805088379594724156782287, 2050582465505075421953614184569, 5368494591426847976106230700218, 14054901308775470617850155894278, 36796209334899567293898859889468, 96333726695923236791786124659030, 252204970752870152025853837879235, 660281185562687233758109413655047, 1728638585935191572665202751553744, 4525634572242887522126561214150246, 11848265130793471055020271612508742, 31019160820137525742129106718131636, 81209217329619106331867692358253415, 212608491168719793513169467267751357, 556616256176540274627836850172490494, 1457240277360901031050232720888332550, 3815104575906162819622949090858609256, 9988073450357587429598593967692209578, 26149115775166599472052900006588835771, 68459273875142210991220152662449828219, 179228705850260033509147671785506995492, 469226843675637889548423023109193035174, 1228451825176653635155861671761940333378, 3216128631854323015951102426811618065048, 8419934070386315412749126317527772185023, 22043673579304623222379897669261546913185, 57711086667527554254525868542601575300770, 151089586423278039541416630954377734158342, 395557672602306564370078249168710889089524
Generating function in Maple syntax:
(2*x^6+2*x^4+2*x^3-8*x^2+5*x-1)/(x^2-3*x+1)/(x^2+x-1)/(x-1)^2
Generating function in latex syntax:
\frac{2 x^{6}+2 x^{4}+2 x^{3}-8 x^{2}+5 x -1}{\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(2*x**6 + 2*x**4 + 2*x**3 - 8*x**2 + 5*x - 1)/((x - 1)**2*(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)^2*F(x)-2*x^6-2*x^4-2*x^3+8*x^2-5*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)^{2} F \! \left(x \right)-2 x^{6}-2 x^{4}-2 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/10*(14*5^(1/2)-25)*(3/2-1/2*5^(1/2))^(-n)+1/10*(-14*5^(1/2)-25)*(3/2+1/2*5^(1/2))^(-n)+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)+5-2*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(14 \sqrt{5}-25\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(-14 \sqrt{5}-25\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\\\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}+5-2 n & \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) = 62
a(6) = 179
a(n+4) = a(n)-2*a(n+1)-3*a(n+2)+4*a(n+3)-2*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) = 20
a \! \left(5\right) = 62
a \! \left(6\right) = 179
a \! \left(n +4\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right)-2 n +5, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15758/
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[16,x]*F[4,x]
F[4,x] = F[24,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[21,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[13,x]+F[17,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = x
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[16,x]*F[20,x]
F[20,x] = F[10,x]
F[21,x] = F[22,x]
F[22,x] = F[16,x]*F[23,x]
F[23,x] = F[9,x]
F[24,x] = F[25,x]+F[41,x]
F[25,x] = F[26,x]
F[26,x] = F[16,x]*F[27,x]
F[27,x] = F[13,x]+F[28,x]
F[28,x] = F[25,x]+F[29,x]
F[29,x] = F[30,x]+F[31,x]+F[35,x]
F[30,x] = 0
F[31,x] = F[16,x]*F[32,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[14,x]
F[34,x] = F[29,x]
F[35,x] = F[16,x]*F[36,x]
F[36,x] = F[25,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[16,x]*F[39,x]
F[39,x] = F[33,x]+F[40,x]
F[40,x] = F[37,x]
F[41,x] = F[30,x]+F[42,x]+F[56,x]
F[42,x] = F[16,x]*F[43,x]
F[43,x] = F[44,x]+F[48,x]
F[44,x] = F[45,x]+F[6,x]
F[45,x] = F[46,x]
F[46,x] = F[16,x]*F[47,x]
F[47,x] = F[6,x]
F[48,x] = F[41,x]+F[49,x]
F[49,x] = 2*F[30,x]+F[50,x]+F[54,x]
F[50,x] = F[16,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[45,x]
F[53,x] = F[49,x]
F[54,x] = F[16,x]*F[55,x]
F[55,x] = F[41,x]
F[56,x] = F[16,x]*F[57,x]
F[57,x] = F[24,x]+F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[38,x]
F[60,x] = F[61,x]
F[61,x] = F[16,x]*F[62,x]
F[62,x] = F[58,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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{24}\! \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_{16}\! \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_{21}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{9}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{41}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{30}\! \left(x \right) = 0
F_{31}\! \left(x \right) = F_{16}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right)
F_{34}\! \left(x \right) = F_{29}\! \left(x \right)
F_{35}\! \left(x \right) = F_{16}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{25}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{16}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{37}\! \left(x \right)
F_{41}\! \left(x \right) = F_{30}\! \left(x \right)+F_{42}\! \left(x \right)+F_{56}\! \left(x \right)
F_{42}\! \left(x \right) = F_{16}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{48}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{6}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{16}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{6}\! \left(x \right)
F_{48}\! \left(x \right) = F_{41}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{50}\! \left(x \right) = F_{16}\! \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_{45}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{16}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{41}\! \left(x \right)
F_{56}\! \left(x \right) = F_{16}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{24}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{38}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{16}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{58}\! \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_16(x)*F_4(x))
Eq(F_4(x), F_24(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_16(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_21(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_13(x) + F_17(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_16(x)*F_20(x))
Eq(F_20(x), F_10(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_16(x)*F_23(x))
Eq(F_23(x), F_9(x))
Eq(F_24(x), F_25(x) + F_41(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_16(x)*F_27(x))
Eq(F_27(x), F_13(x) + F_28(x))
Eq(F_28(x), F_25(x) + F_29(x))
Eq(F_29(x), F_30(x) + F_31(x) + F_35(x))
Eq(F_30(x), 0)
Eq(F_31(x), F_16(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_14(x))
Eq(F_34(x), F_29(x))
Eq(F_35(x), F_16(x)*F_36(x))
Eq(F_36(x), F_25(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_16(x)*F_39(x))
Eq(F_39(x), F_33(x) + F_40(x))
Eq(F_40(x), F_37(x))
Eq(F_41(x), F_30(x) + F_42(x) + F_56(x))
Eq(F_42(x), F_16(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_48(x))
Eq(F_44(x), F_45(x) + F_6(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_16(x)*F_47(x))
Eq(F_47(x), F_6(x))
Eq(F_48(x), F_41(x) + F_49(x))
Eq(F_49(x), 2*F_30(x) + F_50(x) + F_54(x))
Eq(F_50(x), F_16(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_45(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), F_16(x)*F_55(x))
Eq(F_55(x), F_41(x))
Eq(F_56(x), F_16(x)*F_57(x))
Eq(F_57(x), F_24(x) + F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_38(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_16(x)*F_62(x))
Eq(F_62(x), F_58(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, 0, 2, 1], "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, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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