0231_1230_1302_2013_3012
Counting sequence:
1, 1, 2, 6, 19, 55, 148, 381, 957, 2375, 5871, 14533, 36141, 90444, 227895, 578067, 1475142, 3783914, 9748213, 25201982, 65337700, 169767806, 441875012, 1151670746, 3004758024, 7845837335, 20499212848, 53584707996, 140120711717, 366509012985, 958869480946, 2509025716363, 6566060216951, 17184860004209, 44979929903125, 117737749883547, 308198960062767, 806790410888254, 2112034833715973, 5529039212428017, 14474533047838000, 37893460419551236, 99203649187663147, 259713089097040836, 679926822010562454, 1780049784748739840, 4660187347863719298, 12200441890098405736, 31940997584943323202, 83622269389755050145, 218925247634368620246, 573152347613444200818, 1500529543406150549511, 3928431779005380350699, 10284756786410736057280, 26925820565828318658377, 70492668882277201298017, 184552114023409247678219, 483163529072762466279083, 1264938184964501999874977, 3311650449359991230381025, 8670012010193967084912416, 22698383275378900811187907, 59425133204256716921825151, 155577007114019213100111434, 407305869691056848669597070, 1066340565065663185490257873, 2791715751718956412963695226, 7308806542517253463725185912, 19134703580684898798859855730, 50095303609241632574149598896, 131151206066448378206178559398, 343358312228920260609564449196, 898923725897945920752870608539, 2353412856020184535909758042860, 6161314823273141755497823821432, 16130531576020308867626552933017, 42230279829229921121467512847309, 110560307760553727044947340129918, 289450643150199805109717215298487, 757791621085582778476889719921275, 1983924218897622710706322771347221, 5193981033189433714412820246378417, 13598018875834975154073621270719191, 35600075584644085190891010169974339, 93202207858754467304765342445841650, 244006547952933690495736883578985073, 638817435922675351727109041112049341, 1672445759660349859774917705397005972, 4378519842748889217776299006356527240, 11463113767967347773911289176225463271, 30010821459915214064672188247423301728, 78569350609302414341534515016248874506, 205697230363040268802789835701729627604, 538522340469914871752551949889749942990, 1409869791026897305826299929569137272212, 3691087032571162964469215670020893091142, 9663391306607363425067082742900001382149, 25299086887092470985703503883492026628978, 66233869354353136881986371557201906325422, 173402521175333114360141496087365344518875
Generating function in Maple syntax:
(2*x^8-11*x^7+28*x^6-58*x^5+81*x^4-69*x^3+34*x^2-9*x+1)/(2*x-1)/(x^2-3*x+1)/(x-1)^5
Generating function in latex syntax:
\frac{2 x^{8}-11 x^{7}+28 x^{6}-58 x^{5}+81 x^{4}-69 x^{3}+34 x^{2}-9 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(2*x**8 - 11*x**7 + 28*x**6 - 58*x**5 + 81*x**4 - 69*x**3 + 34*x**2 - 9*x + 1)/((x - 1)**5*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^5*F(x)-2*x^8+11*x^7-28*x^6+58*x^5-81*x^4+69*x^3-34*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{5} F \! \left(x \right)-2 x^{8}+11 x^{7}-28 x^{6}+58 x^{5}-81 x^{4}+69 x^{3}-34 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/120*(-12*5^(1/2)+60)*(3/2-1/2*5^(1/2))^(-n)+1/120*(12*5^(1/2)+60)*(3/2+1/2*5^(1/2))^(-n)-1/24*n^4+1/4*n^3-47/24*n^2+3/4*n+2^(n+1)-3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-12 \sqrt{5}+60\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(12 \sqrt{5}+60\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}-\frac{n^{4}}{24}+\frac{n^{3}}{4}-\\\frac{47 n^{2}}{24}+\frac{3 n}{4}+2^{n +1}-3 & \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) = 148
a(7) = 381
a(8) = 957
a(n+3) = -1/24*n^4+1/4*n^3-23/24*n^2-7*a(n+1)+5*a(n+2)+2*a(n)-5/4*n+3, n >= 9
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) = 148
a \! \left(7\right) = 381
a \! \left(8\right) = 957
a \! \left(n +3\right) = -\frac{n^{4}}{24}+\frac{n^{3}}{4}-\frac{23 n^{2}}{24}-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+2 a \! \left(n \right)-\frac{5 n}{4}+3, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21054/
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[13,x]*F[4,x]
F[4,x] = F[25,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = F[1,x]+F[10,x]
F[13,x] = x
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[13,x]*F[17,x]
F[17,x] = F[14,x]+F[6,x]
F[18,x] = F[13,x]*F[19,x]
F[19,x] = F[20,x]+F[21,x]
F[20,x] = F[10,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[13,x]*F[24,x]
F[24,x] = F[10,x]+F[22,x]
F[25,x] = F[26,x]+F[36,x]
F[26,x] = F[27,x]
F[27,x] = F[13,x]*F[28,x]
F[28,x] = F[12,x]+F[29,x]
F[29,x] = F[26,x]+F[30,x]
F[30,x] = F[15,x]+F[31,x]+F[35,x]
F[31,x] = F[13,x]*F[32,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[10,x]
F[34,x] = F[30,x]
F[35,x] = F[13,x]*F[29,x]
F[36,x] = F[15,x]+F[37,x]+F[48,x]
F[37,x] = F[13,x]*F[38,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[40,x]+F[6,x]
F[40,x] = F[41,x]
F[41,x] = F[13,x]*F[39,x]
F[42,x] = F[36,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[13,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[40,x]
F[47,x] = F[43,x]
F[48,x] = F[13,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[26,x]
F[51,x] = F[30,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{25}\! \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_{13}\! \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_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \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_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{36}\! \left(x \right)
F_{26}\! \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_{12}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \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_{10}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{48}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{6}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right) F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{36}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{13}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{40}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{13}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{26}\! \left(x \right)
F_{51}\! \left(x \right) = F_{30}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_25(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_13(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), F_1(x) + F_10(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_13(x)*F_17(x))
Eq(F_17(x), F_14(x) + F_6(x))
Eq(F_18(x), F_13(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_21(x))
Eq(F_20(x), F_10(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_13(x)*F_24(x))
Eq(F_24(x), F_10(x) + F_22(x))
Eq(F_25(x), F_26(x) + F_36(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_13(x)*F_28(x))
Eq(F_28(x), F_12(x) + F_29(x))
Eq(F_29(x), F_26(x) + F_30(x))
Eq(F_30(x), F_15(x) + F_31(x) + F_35(x))
Eq(F_31(x), F_13(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_10(x))
Eq(F_34(x), F_30(x))
Eq(F_35(x), F_13(x)*F_29(x))
Eq(F_36(x), F_15(x) + F_37(x) + F_48(x))
Eq(F_37(x), F_13(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_40(x) + F_6(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_13(x)*F_39(x))
Eq(F_42(x), F_36(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_13(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_40(x))
Eq(F_47(x), F_43(x))
Eq(F_48(x), F_13(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_26(x))
Eq(F_51(x), F_30(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, 3, 1], "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": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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, 3, 1], "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": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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