0132_0213_1320_3120
Counting sequence:
1, 1, 2, 6, 20, 64, 190, 530, 1416, 3681, 9417, 23892, 60419, 152774, 386989, 983033, 2505310, 6406584, 16436712, 42298294, 109148570, 282330677, 731815667, 1900273016, 4941742405, 12867284794, 33538479735, 87492304765, 228401566896, 596588326076, 1559014605122, 4075556546682, 10657454492524, 27875600535289, 72925323704509, 190809102529708, 499313005330695, 1306739071446174, 3420096755164577, 8951884746745809, 23432121511245890, 61337401680898672, 160565515002393532, 420329181634437326, 1100360457249776190, 2880625746277711149, 7541257296839217255, 19742613980612113200, 51685493929462385801, 135311633600147421746, 354244832902608350827, 927413506064702022197, 2427976544993081413204, 6356477003892779459604, 16641374527791871164054, 43567483323996341878258, 114060742177824729081680, 298614063165934112451089, 781780060211292378189617, 2046723289207377033478724, 5358384042803315688045835, 14028417093814741848441142, 36726843315519689265240597, 96152064141810756308035561, 251729249958663183470067750, 659035483972915487903990504, 1725376791520027640204426000, 4517094055871998097352124710, 11825903678995511870573249506, 30960613531573395730681523941, 81055929905961927311841762907, 212207161945425961301069974312, 555565527005821249014791285709, 1454489360337604655052091599594, 3807902434767239023682941931999, 9969217701942830168927417061933, 26099750179935137264656207099640, 68330031841443253739546064220204, 178890343323221769285773834175082, 468340994029207946986923392895914, 1226132630453037061824420768489780, 3210056880480499627607444790191337, 8404037976836307416884639416723237, 22002056980817419450108953208141596, 57602132825380555858145235941880335, 150804341211226680514888898561646734, 394810890232850795549957972583022345, 1033628328321921216026482494770772545, 2706074092373089652641733362702949386, 7084593944019672902781703094899381536, 18547707730014522498786342524345904612, 48558529226448967722377248881296051902, 127127879909718299411213235322770401414, 332825110422549312967534084287296988509, 871347451195759494345004201527335424719, 2281217242836675059656852747446441504608, 5972304276650729823568590213466058647825, 15635695585773587408463565674961083927394, 40934782477956467835708553248838312987347, 107168651842609265844549288693200936630629, 280571173038779386945942305567668344019100
Generating function in Maple syntax:
(x^9-4*x^8-3*x^7+42*x^6-98*x^5+125*x^4-95*x^3+42*x^2-10*x+1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
\frac{x^{9}-4 x^{8}-3 x^{7}+42 x^{6}-98 x^{5}+125 x^{4}-95 x^{3}+42 x^{2}-10 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**9 - 4*x**8 - 3*x**7 + 42*x**6 - 98*x**5 + 125*x**4 - 95*x**3 + 42*x**2 - 10*x + 1)/((x - 1)**4*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(2*x-1)^2*(x-1)^4*F(x)+x^9-4*x^8-3*x^7+42*x^6-98*x^5+125*x^4-95*x^3+42*x^2-10*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+x^{9}-4 x^{8}-3 x^{7}+42 x^{6}-98 x^{5}+125 x^{4}-95 x^{3}+42 x^{2}-10 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,-1/6*n^3+3/8*2^n*n-2^n+1+1/6*n-1/5*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)+1/5*(3/2-1/2*5^(1/2))^(-n)*5^(1/2))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -\frac{n^{3}}{6}+\frac{3 \,2^{n} n}{8}-2^{n}+1+\frac{n}{6}-\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+\frac{\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5} & \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) = 64
a(6) = 190
a(7) = 530
a(8) = 1416
a(9) = 3681
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)+1/6*n*(n^2-3*n-16), n >= 10
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) = 64
a \! \left(6\right) = 190
a \! \left(7\right) = 530
a \! \left(8\right) = 1416
a \! \left(9\right) = 3681
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)+\frac{n \left(n^{2}-3 n -16\right)}{6}, \quad n \geq 10
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/21200/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[15,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[43,x]
F[4,x] = F[15,x]*F[5,x]
F[5,x] = F[1,x]+F[29,x]+F[30,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[15,x]*F[19,x]*F[8,x]
F[8,x] = F[1,x]+F[4,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[15,x]
F[11,x] = F[12,x]+F[22,x]
F[12,x] = F[13,x]*F[16,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[13,x]*F[15,x]
F[15,x] = x
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[13,x]*F[15,x]*F[19,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[15,x]*F[19,x]
F[22,x] = F[20,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[26,x]
F[24,x] = 0
F[25,x] = F[15,x]*F[22,x]
F[26,x] = F[15,x]*F[27,x]
F[27,x] = F[23,x]+F[28,x]
F[28,x] = F[14,x]
F[29,x] = F[15,x]*F[5,x]
F[30,x] = F[15,x]*F[31,x]
F[31,x] = F[1,x]+F[30,x]+F[32,x]+F[36,x]+F[38,x]
F[32,x] = F[33,x]
F[33,x] = F[15,x]*F[19,x]*F[34,x]
F[34,x] = F[1,x]+F[32,x]+F[35,x]
F[35,x] = F[15,x]*F[34,x]
F[36,x] = F[15,x]*F[37,x]
F[37,x] = 2*F[36,x]+F[1,x]+F[32,x]
F[38,x] = F[39,x]
F[39,x] = F[13,x]^2*F[15,x]*F[40,x]
F[40,x] = F[1,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[15,x]*F[19,x]*F[40,x]
F[43,x] = F[15,x]*F[44,x]
F[44,x] = F[1,x]+F[43,x]+F[45,x]+F[9,x]
F[45,x] = F[15,x]*F[46,x]
F[46,x] = F[1,x]+F[47,x]+F[48,x]+F[51,x]+F[6,x]
F[47,x] = F[15,x]*F[46,x]
F[48,x] = F[49,x]
F[49,x] = F[15,x]*F[40,x]*F[50,x]
F[50,x] = F[19,x]+F[27,x]
F[51,x] = F[15,x]*F[52,x]
F[52,x] = F[1,x]+F[32,x]+F[36,x]+F[51,x]+F[53,x]+F[55,x]
F[53,x] = F[54,x]
F[54,x] = F[13,x]*F[15,x]*F[40,x]
F[55,x] = F[56,x]
F[56,x] = F[13,x]^2*F[15,x]*F[40,x]*F[57,x]
F[57,x] = F[1,x]+F[15,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_{15}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{43}\! \left(x \right)
F_{4}\! \left(x \right) = F_{15}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{15}\! \left(x \right) F_{19}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{22}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right) F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{15}\! \left(x \right) F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right) F_{5}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right) F_{19}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right) F_{34}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{36}\! \left(x \right)+F_{1}\! \left(x \right)+F_{32}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{13} \left(x \right)^{2} F_{15}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right) F_{19}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{1}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)+F_{9}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{1}\! \left(x \right)+F_{47}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)+F_{6}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right) F_{46}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right) F_{40}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{19}\! \left(x \right)+F_{27}\! \left(x \right)
F_{51}\! \left(x \right) = F_{15}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{40}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{13} \left(x \right)^{2} F_{15}\! \left(x \right) F_{40}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \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_15(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_43(x))
Eq(F_4(x), F_15(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_29(x) + F_30(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_15(x)*F_19(x)*F_8(x))
Eq(F_8(x), F_1(x) + F_4(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_15(x))
Eq(F_11(x), F_12(x) + F_22(x))
Eq(F_12(x), F_13(x)*F_16(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_13(x)*F_15(x))
Eq(F_15(x), x)
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_13(x)*F_15(x)*F_19(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_15(x)*F_19(x))
Eq(F_22(x), F_20(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_26(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_15(x)*F_22(x))
Eq(F_26(x), F_15(x)*F_27(x))
Eq(F_27(x), F_23(x) + F_28(x))
Eq(F_28(x), F_14(x))
Eq(F_29(x), F_15(x)*F_5(x))
Eq(F_30(x), F_15(x)*F_31(x))
Eq(F_31(x), F_1(x) + F_30(x) + F_32(x) + F_36(x) + F_38(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_15(x)*F_19(x)*F_34(x))
Eq(F_34(x), F_1(x) + F_32(x) + F_35(x))
Eq(F_35(x), F_15(x)*F_34(x))
Eq(F_36(x), F_15(x)*F_37(x))
Eq(F_37(x), F_1(x) + F_32(x) + 2*F_36(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_13(x)**2*F_15(x)*F_40(x))
Eq(F_40(x), F_1(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_15(x)*F_19(x)*F_40(x))
Eq(F_43(x), F_15(x)*F_44(x))
Eq(F_44(x), F_1(x) + F_43(x) + F_45(x) + F_9(x))
Eq(F_45(x), F_15(x)*F_46(x))
Eq(F_46(x), F_1(x) + F_47(x) + F_48(x) + F_51(x) + F_6(x))
Eq(F_47(x), F_15(x)*F_46(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_15(x)*F_40(x)*F_50(x))
Eq(F_50(x), F_19(x) + F_27(x))
Eq(F_51(x), F_15(x)*F_52(x))
Eq(F_52(x), F_1(x) + F_32(x) + F_36(x) + F_51(x) + F_53(x) + F_55(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_13(x)*F_15(x)*F_40(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_13(x)**2*F_15(x)*F_40(x)*F_57(x))
Eq(F_57(x), F_1(x) + F_15(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, 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