0132_0231_2013_2301_3120
Counting sequence:
1, 1, 2, 6, 19, 52, 124, 267, 535, 1021, 1890, 3444, 6251, 11402, 21024, 39305, 74543, 143243, 278310, 545434, 1075859, 2131920, 4238116, 8443255, 16844743, 33637161, 67209418, 134339056, 268580859, 537044070, 1073946824, 2147725013, 4295250015, 8590264151, 17180251598, 34360180214, 68719985187, 137439536332, 274878572684, 549756571683, 1099512487543, 2199024227989, 4398047607730, 8796094255404, 17592187427467, 35184373635970, 70368745904112, 140737490277345, 281474978845583, 562949955787619, 1125899909459958, 2251799816574482, 4503599630553779, 9007199258241800, 18014398513325172, 36028797023175823, 72057594042536231, 144115188080889921, 288230376157202458, 576460752309403432, 1152921504613350427, 2305843009220756958, 4611686018435048344, 9223372036863073453, 18446744073718528191, 36893488147428802479, 73786976294848674206, 147573952589687697134, 295147905179364976707, 590295810358718721668, 1180591620717425347292, 2361183241434837681851, 4722366482869661379543, 9444732965739307746349, 18889465931478599391746, 37778931862957181532132, 75557863725914344597675, 151115727451828669446010, 302231454903657317789632, 604462909807314613050681, 1208925819614629202070511, 2417851639229258378528827, 4835703278458516729781958, 9671406556917033430539402, 19342813113834066830216915, 38685626227668133627642688, 77371252455336267220469700, 154742504910672534404000423, 309485009821345068768836231, 618970019642690137496176217, 1237940039285380274948414826, 2475880078570760549850337120, 4951760157141521099651509307, 9903520314283042199251059798, 19807040628566084398447241320, 39614081257132168796836555141, 79228162514264337593611999519, 158456325028528675187159566599, 316912650057057350374251236206, 633825300114114700748430963430, 1267650600228229401496786653795
Generating function in Maple syntax:
(2*x^7-8*x^6+19*x^5-30*x^4+33*x^3-21*x^2+7*x-1)/(2*x-1)/(-1+x)^6
Generating function in latex syntax:
\frac{2 x^{7}-8 x^{6}+19 x^{5}-30 x^{4}+33 x^{3}-21 x^{2}+7 x -1}{\left(2 x -1\right) \left(-1+x \right)^{6}}
Generating function in sympy syntax:
(2*x**7 - 8*x**6 + 19*x**5 - 30*x**4 + 33*x**3 - 21*x**2 + 7*x - 1)/((x - 1)**6*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(-1+x)^6*F(x)-2*x^7+8*x^6-19*x^5+30*x^4-33*x^3+21*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(-1+x \right)^{6} F \! \left(x \right)-2 x^{7}+8 x^{6}-19 x^{5}+30 x^{4}-33 x^{3}+21 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1+2^n+1/120*n^5+1/8*n^3-n^2+13/15*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -1+2^{n}+\frac{n^{5}}{120}+\frac{n^{3}}{8}-n^{2}+\frac{13 n}{15} & \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) = 52
a(6) = 124
a(7) = 267
a(n+1) = 2*a(n)-1/120*(n-1)*(n^4-4*n^3+n^2-174*n+120), 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) = 19
a \! \left(5\right) = 52
a \! \left(6\right) = 124
a \! \left(7\right) = 267
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{4}-4 n^{3}+n^{2}-174 n +120\right)}{120}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19202/
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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[11,x]
F[18,x] = F[14,x]
F[19,x] = F[20,x]+F[29,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[6,x]
F[23,x] = F[20,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[7,x]
F[28,x] = F[24,x]
F[29,x] = F[30,x]+F[31,x]+F[48,x]
F[30,x] = 0
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[42,x]
F[33,x] = F[34,x]+F[7,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[41,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[1,x]+F[38,x]
F[41,x] = F[35,x]
F[42,x] = F[29,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[34,x]
F[47,x] = F[43,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[55,x]
F[50,x] = F[38,x]+F[51,x]
F[51,x] = F[30,x]+F[52,x]+F[54,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[11,x]+F[51,x]
F[54,x] = F[4,x]*F[50,x]
F[55,x] = F[56,x]+F[72,x]
F[56,x] = F[30,x]+F[57,x]+F[60,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[7,x]
F[59,x] = F[57,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[66,x]
F[62,x] = F[38,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[11,x]+F[63,x]
F[66,x] = F[56,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[63,x]
F[71,x] = F[67,x]
F[72,x] = F[73,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[75,x]+F[76,x]
F[75,x] = F[51,x]
F[76,x] = F[72,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_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{29}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \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_{6}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{24}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{48}\! \left(x \right)
F_{30}\! \left(x \right) = 0
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{42}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{41}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{35}\! \left(x \right)
F_{42}\! \left(x \right) = F_{29}\! \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_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{34}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{55}\! \left(x \right)
F_{50}\! \left(x \right) = F_{38}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{30}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \left(x \right)+F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{72}\! \left(x \right)
F_{56}\! \left(x \right) = F_{30}\! \left(x \right)+F_{57}\! \left(x \right)+F_{60}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{7}\! \left(x \right)
F_{59}\! \left(x \right) = F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{66}\! \left(x \right)
F_{62}\! \left(x \right) = F_{38}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{11}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{56}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{63}\! \left(x \right)
F_{71}\! \left(x \right) = F_{67}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{76}\! \left(x \right)
F_{75}\! \left(x \right) = F_{51}\! \left(x \right)
F_{76}\! \left(x \right) = F_{72}\! \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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_20(x) + F_29(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_6(x))
Eq(F_23(x), F_20(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_7(x))
Eq(F_28(x), F_24(x))
Eq(F_29(x), F_30(x) + F_31(x) + F_48(x))
Eq(F_30(x), 0)
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_42(x))
Eq(F_33(x), F_34(x) + F_7(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_41(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_1(x) + F_38(x))
Eq(F_41(x), F_35(x))
Eq(F_42(x), F_29(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_46(x) + F_47(x))
Eq(F_46(x), F_34(x))
Eq(F_47(x), F_43(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_55(x))
Eq(F_50(x), F_38(x) + F_51(x))
Eq(F_51(x), F_30(x) + F_52(x) + F_54(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_11(x) + F_51(x))
Eq(F_54(x), F_4(x)*F_50(x))
Eq(F_55(x), F_56(x) + F_72(x))
Eq(F_56(x), F_30(x) + F_57(x) + F_60(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_7(x))
Eq(F_59(x), F_57(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_66(x))
Eq(F_62(x), F_38(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_11(x) + F_63(x))
Eq(F_66(x), F_56(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_63(x))
Eq(F_71(x), F_67(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_76(x))
Eq(F_75(x), F_51(x))
Eq(F_76(x), F_72(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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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