0123_0132_1203_2031_2130
Counting sequence:
1, 1, 2, 6, 19, 55, 150, 392, 996, 2485, 6128, 14999, 36536, 88723, 215021, 520425, 1258530, 3041754, 7348905, 17750627, 42868003, 103515440, 249945419, 603481490, 1457029999, 3517738134, 8492824328, 20503901292, 49501459249, 119508166381, 288519970668, 696551632670, 1681628939298, 4059818739163, 9801281348438, 23662405594349, 57126131625830, 137914732092553, 332955698145683, 803826293964687, 1940608553990016, 4685043835439856, 11310696926279199, 27306438822902201, 65923576408396321, 159153594610910774, 384230770437745745, 927615143265145256, 2239461069554306269, 5406537302738769900, 13052535707983127246, 31511608772021316690, 76075753338293333071, 183663115588191846499, 443401984740528461058, 1070467085434684066100, 2584336156201183324512, 6239139398793772742593, 15062614955336737567100, 36364369311971978660291, 87791353583333434427156, 211947076485196317836095, 511685506564336279958777, 1235318089631036557932981, 2982321685854187285861854, 7199961461384356699871430, 17382244608695624145854805, 41964450678893274022044047, 101311145966672564680654111, 244586742612546464904524540, 590484631192263948501584663, 1425556004997880877440745534, 3441596641189330672928006731, 8308749287378653708374739458, 20059095215950054544300394836, 48426939719284290736676416440, 116912974654527580411977021805, 282252889028353923894655138969, 681418752711258844929635770200, 1645090394450909502816299826122, 3971599541613139156352957036958, 9588289477677287010375308658535, 23148178496967873677747390724194, 55884646471613294061367001232665, 134917471440194881996622120682434, 325719589352003737946248881213109, 786356650144203457976898249213983, 1898432889640412433879461384358747, 4583222429425031205803015388751164, 11064877748490479505532102537395036, 26712977926405997756981025209891403, 64490833601302487219654568079058405, 155694645129010991936564381236235293, 375880123859324503033217965541632898, 907454892847660049683709167177828253, 2190789909554644686021779789745716564, 5289034711956949557029121091376011705, 12768859333468544019003017807052913368, 30826753378894037949260004884743757978, 74422366091256620490670871590357517982, 179671485561407279857974440258537797771
Generating function in Maple syntax:
(x^8-2*x^6+5*x^4+x^3-8*x^2+5*x-1)/(x^2+x-1)/(x^2+2*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{8}-2 x^{6}+5 x^{4}+x^{3}-8 x^{2}+5 x -1}{\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**8 - 2*x**6 + 5*x**4 + x**3 - 8*x**2 + 5*x - 1)/((x - 1)**3*(x**2 + x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+x-1)*(x^2+2*x-1)*(x-1)^3*F(x)+x^8-2*x^6+5*x^4+x^3-8*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{8}-2 x^{6}+5 x^{4}+x^{3}-8 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,2,1/40*(16*5^(1/2)-40)*(-1/2*5^(1/2)-1/2)^(-n)+1/40*(-16*5^(1/2)-40)*(1/2*5^(1/2)-1/2)^(-n)+1/40*(5*2^(1/2)+45)*(-1-2^(1/2))^(-n)+1/40*(-5*2^(1/2)+45)*(2^(1/2)-1)^(-n)-1/4*n^2+3/4*n+3/4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}2 & n =0 \\ \frac{\left(16 \sqrt{5}-40\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{40}+\frac{\left(-16 \sqrt{5}-40\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{40}+\\\frac{\left(5 \sqrt{2}+45\right) \left(-1-\sqrt{2}\right)^{-n}}{40}+\frac{\left(-5 \sqrt{2}+45\right) \left(\sqrt{2}-1\right)^{-n}}{40}-\frac{n^{2}}{4}+\frac{3 n}{4}+\frac{3}{4} & \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) = 150
a(7) = 392
a(8) = 996
a(n+1) = -1/6*n^2-1/3*a(n)+a(n+3)-1/3*a(n+4)+5/6*n+2/3, 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) = 150
a \! \left(7\right) = 392
a \! \left(8\right) = 996
a \! \left(n +1\right) = -\frac{n^{2}}{6}-\frac{a \! \left(n \right)}{3}+a \! \left(n +3\right)-\frac{a \! \left(n +4\right)}{3}+\frac{5 n}{6}+\frac{2}{3}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16441/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[14,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[32,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[25,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[14,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[21,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[24,x]
F[31,x] = F[28,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[2,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[2,x]*F[4,x]
F[37,x] = F[20,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[20,x]*F[4,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{32}\! \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_{25}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{24}\! \left(x \right)
F_{31}\! \left(x \right) = F_{28}\! \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_{37}\! \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_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{20}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_32(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_25(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_14(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_21(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_24(x))
Eq(F_31(x), F_28(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_2(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_2(x)*F_4(x))
Eq(F_37(x), F_20(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_20(x)*F_4(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 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