0123_0132_1023_1302_2301
Counting sequence:
1, 1, 2, 6, 19, 52, 123, 261, 514, 965, 1764, 3192, 5789, 10610, 19737, 37303, 71540, 138875, 272122, 536866, 1064231, 2116416, 4217767, 8416921, 16811094, 33594657, 67156288, 134273276, 268500129, 536945790, 1073828069, 2147582507, 4295080104, 8590062775, 17180014262, 34359901958, 68719660555, 137439159340, 274878136787, 549756069741, 1099511911786, 2199023569981, 4398046858332, 8796093404736, 17592186464869, 35184372549962, 70368744682353, 140737488906591, 281474977311644, 562949954075315, 1125899907553074, 2251799814455722, 4503599628204719, 9007199255642840, 18014398510455487, 36028797020013313, 72057594039057470, 144115188077070105, 288230376153015352, 576460752304821316, 1152921504608344041, 2305843009215295446, 4611686018429099197, 9223372036856602451, 18446744073711499344, 36893488147421177967, 73786976294840414318, 147573952589678760206, 295147905179355319059, 590295810358708297540, 1180591620717414108779, 2361183241434825578837, 4722366482869648359634, 9444732965739293754805, 18889465931478584371412, 37778931862957165423368, 75557863725914327338285, 151115727451828650971170, 302231454903657298031817, 604462909807314591939591, 1208925819614629179532996, 2417851639229258354488811, 4835703278458516704160362, 9671406556917033403254066, 19342813113834066801182519, 38685626227668133596770672, 77371252455336267187668183, 154742504910672534369174121, 309485009821345068731886374, 618970019642690137457000465, 1237940039285380274906907184, 2475880078570760549806387852, 4951760157141521099605004849, 9903520314283042199201882670, 19807040628566084398395270037, 39614081257132168796781664123, 79228162514264337593554059000, 158456325028528675187098442535, 316912650057057350374186790182, 633825300114114700748363052566, 1267650600228229401496715130651
Generating function in Maple syntax:
-(7*x^5-12*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{7 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-7*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)+7*x^5-12*x^4+18*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+7 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
7/8*n^2-17/12*n-7/12*n^3+1/8*n^4+2^n
Explicit closed form in latex syntax:
\frac{7 n^{2}}{8}-\frac{17 n}{12}-\frac{7 n^{3}}{12}+\frac{n^{4}}{8}+2^{n}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 52
a(n+1) = 2*a(n)-1/24*(n-1)*(3*n^3-23*n^2+22*n-24), n >= 6
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(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(3 n^{3}-23 n^{2}+22 n -24\right)}{24}, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16314/
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[10,x]*F[4,x]
F[4,x] = F[22,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[11,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = x
F[11,x] = F[12,x]+F[6,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[10,x]*F[15,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[10,x]
F[17,x] = F[12,x]
F[18,x] = F[10,x]*F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[21,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[13,x]+F[24,x]+F[46,x]
F[24,x] = F[10,x]*F[25,x]
F[25,x] = F[26,x]+F[31,x]
F[26,x] = F[19,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[10,x]*F[29,x]
F[29,x] = F[16,x]+F[30,x]
F[30,x] = F[28,x]
F[31,x] = F[32,x]+F[38,x]
F[32,x] = F[13,x]+F[24,x]+F[33,x]
F[33,x] = F[10,x]*F[34,x]
F[34,x] = F[19,x]+F[35,x]
F[35,x] = F[13,x]+F[33,x]+F[36,x]
F[36,x] = F[10,x]*F[37,x]
F[37,x] = F[19,x]+F[35,x]
F[38,x] = F[39,x]
F[39,x] = F[10,x]*F[40,x]
F[40,x] = F[41,x]+F[45,x]
F[41,x] = F[42,x]
F[42,x] = F[13,x]+F[18,x]+F[43,x]
F[43,x] = F[10,x]*F[44,x]
F[44,x] = F[10,x]+F[42,x]
F[45,x] = F[39,x]
F[46,x] = F[10,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[19,x]+F[42,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[13,x]+F[36,x]+F[46,x]
F[51,x] = F[13,x]+F[52,x]+F[53,x]+F[56,x]
F[52,x] = 0
F[53,x] = F[10,x]*F[54,x]
F[54,x] = F[41,x]+F[55,x]
F[55,x] = F[51,x]
F[56,x] = F[10,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[10,x]*F[59,x]
F[59,x] = F[19,x]+F[57,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{22}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{11}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{6}\! \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_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right)+F_{24}\! \left(x \right)+F_{46}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = F_{19}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{38}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right)+F_{24}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{19}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right)+F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{19}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right)+F_{18}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{39}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{19}\! \left(x \right)+F_{42}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \left(x \right)+F_{36}\! \left(x \right)+F_{46}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right)+F_{52}\! \left(x \right)+F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{52}\! \left(x \right) = 0
F_{53}\! \left(x \right) = F_{10}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{41}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{10}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{19}\! \left(x \right)+F_{57}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_22(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_10(x)*F_8(x))
Eq(F_8(x), F_11(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_6(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_10(x)*F_15(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_10(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_10(x)*F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_10(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_13(x) + F_24(x) + F_46(x))
Eq(F_24(x), F_10(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_31(x))
Eq(F_26(x), F_19(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_10(x)*F_29(x))
Eq(F_29(x), F_16(x) + F_30(x))
Eq(F_30(x), F_28(x))
Eq(F_31(x), F_32(x) + F_38(x))
Eq(F_32(x), F_13(x) + F_24(x) + F_33(x))
Eq(F_33(x), F_10(x)*F_34(x))
Eq(F_34(x), F_19(x) + F_35(x))
Eq(F_35(x), F_13(x) + F_33(x) + F_36(x))
Eq(F_36(x), F_10(x)*F_37(x))
Eq(F_37(x), F_19(x) + F_35(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_10(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_45(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_13(x) + F_18(x) + F_43(x))
Eq(F_43(x), F_10(x)*F_44(x))
Eq(F_44(x), F_10(x) + F_42(x))
Eq(F_45(x), F_39(x))
Eq(F_46(x), F_10(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_19(x) + F_42(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_13(x) + F_36(x) + F_46(x))
Eq(F_51(x), F_13(x) + F_52(x) + F_53(x) + F_56(x))
Eq(F_52(x), 0)
Eq(F_53(x), F_10(x)*F_54(x))
Eq(F_54(x), F_41(x) + F_55(x))
Eq(F_55(x), F_51(x))
Eq(F_56(x), F_10(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_10(x)*F_59(x))
Eq(F_59(x), F_19(x) + F_57(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, 0, 2, 3], "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, 3, 0, 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, 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, 0, 2, 3], "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, 3, 0, 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