0123_0231_1203_1230_2301
Counting sequence:
1, 1, 2, 6, 19, 56, 150, 369, 849, 1859, 3930, 8112, 16487, 33194, 66478, 132787, 264965, 528637, 1054978, 2106250, 4206875, 8405580, 16799686, 33583685, 67146393, 134265255, 268494954, 536944628, 1073832303, 2147593758, 4295100254, 8590093991, 17180059021, 34359963073, 68719741202, 137439263086, 274878267619, 549756232096, 1099512110582, 2199023810649, 4398047146849, 8796093747659, 17592186869370, 35184373023864, 70368745234167, 140737489545554, 281474978047758, 562949954919387, 1125899908516757, 2251799815551557, 4503599629446178, 9007199257044370, 18014398512032555, 36028797021782452, 72057594041036326, 144115188079277485, 288230376155471273, 576460752307547055, 1152921504611362186, 2305843009218629948, 4611686018432775423, 9223372036860647238, 18446744073715941054, 36893488147426046543, 73786976294845741341, 147573952589684578953, 295147905179361664562, 590295810358715206646, 1180591620717421620211, 2361183241434833733256, 4722366482869657199702, 9444732965739303325249, 18889465931478594719089, 37778931862957176597331, 75557863725914339389850, 151115727451828663953984, 302231454903657312001927, 604462909807314606955514, 1208925819614629195655790, 2417851639229258371782147, 4835703278458516722690597, 9671406556917033423090317, 19342813113834066822396738, 38685626227668133619437722, 77371252455336267211865915, 154742504910672534394983452, 309485009821345068759391366, 618970019642690137486288405, 1237940039285380274938068665, 2475880078570760549839516855, 4951760157141521099640198826, 9903520314283042199239242628, 19807040628566084398434900623, 39614081257132168796823673710, 79228162514264337593598559774, 158456325028528675187145550583, 316912650057057350374236625581, 633825300114114700748415739473, 1267650600228229401496770797394, 2535301200456458802993477605246, 5070602400912917605986887770371
Generating function in Maple syntax:
-(2*x^6-15*x^5+30*x^4-33*x^3+21*x^2-7*x+1)/(2*x-1)/(x-1)^6
Generating function in latex syntax:
-\frac{2 x^{6}-15 x^{5}+30 x^{4}-33 x^{3}+21 x^{2}-7 x +1}{\left(2 x -1\right) \left(x -1\right)^{6}}
Generating function in sympy syntax:
(-2*x**6 + 15*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)*(x-1)^6*F(x)+2*x^6-15*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(x -1\right)^{6} F \! \left(x \right)+2 x^{6}-15 x^{5}+30 x^{4}-33 x^{3}+21 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
4*2^n-89/30*n+1/120*n^5-1/12*n^4-1/24*n^3-11/12*n^2-3
Explicit closed form in latex syntax:
4 \,2^{n}-\frac{89 n}{30}+\frac{n^{5}}{120}-\frac{n^{4}}{12}-\frac{n^{3}}{24}-\frac{11 n^{2}}{12}-3
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 150
a(n+1) = 2*a(n)-1/120*(n-1)*(n^4-14*n^3+11*n^2-34*n-120), n >= 7
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) = 56
a \! \left(6\right) = 150
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{4}-14 n^{3}+11 n^{2}-34 n -120\right)}{120}, \quad n \geq 7
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21533/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[12,x]*F[3,x]
F[3,x] = F[28,x]+F[4,x]
F[4,x] = F[1,x]+F[5,x]
F[5,x] = F[12,x]*F[6,x]
F[6,x] = F[4,x]+F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[7,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[15,x]*F[9,x]
F[15,x] = F[16,x]+F[23,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[19,x]+F[20,x]+F[22,x]
F[19,x] = 0
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[10,x]+F[18,x]
F[22,x] = F[12,x]*F[17,x]
F[23,x] = F[10,x]+F[24,x]
F[24,x] = F[19,x]+F[25,x]+F[26,x]
F[25,x] = F[12,x]*F[23,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[10,x]+F[24,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[58,x]+F[60,x]
F[31,x] = F[32,x]+F[9,x]
F[32,x] = F[33,x]+F[42,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[9,x]
F[36,x] = F[33,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[10,x]
F[41,x] = F[37,x]
F[42,x] = F[19,x]+F[22,x]+F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[52,x]
F[45,x] = F[10,x]+F[46,x]
F[46,x] = F[19,x]+F[47,x]+F[50,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[10,x]+F[49,x]
F[49,x] = F[47,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[10,x]+F[46,x]
F[52,x] = F[42,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[46,x]
F[57,x] = F[53,x]
F[58,x] = F[59,x]
F[59,x] = F[9,x]^2*F[12,x]*F[16,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[30,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[12,x]*F[64,x]*F[65,x]*F[9,x]
F[64,x] = F[27,x]+F[9,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[9,x]^2
F[67,x] = F[68,x]
F[68,x] = F[12,x]*F[51,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_{12}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{28}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{1}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{12}\! \left(x \right) F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{4}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{7}\! \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) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{9}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{23}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)+F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{9}\! \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_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{9}\! \left(x \right)
F_{36}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{10}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)
F_{42}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{52}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{19}\! \left(x \right)+F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{10}\! \left(x \right)+F_{46}\! \left(x \right)
F_{52}\! \left(x \right) = F_{42}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{46}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{30}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right) F_{65}\! \left(x \right) F_{9}\! \left(x \right)
F_{64}\! \left(x \right) = F_{27}\! \left(x \right)+F_{9}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{9} \left(x \right)^{2}
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \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_12(x)*F_3(x))
Eq(F_3(x), F_28(x) + F_4(x))
Eq(F_4(x), F_1(x) + F_5(x))
Eq(F_5(x), F_12(x)*F_6(x))
Eq(F_6(x), F_4(x) + F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_7(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_15(x)*F_9(x))
Eq(F_15(x), F_16(x) + F_23(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_22(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_10(x) + F_18(x))
Eq(F_22(x), F_12(x)*F_17(x))
Eq(F_23(x), F_10(x) + F_24(x))
Eq(F_24(x), F_19(x) + F_25(x) + F_26(x))
Eq(F_25(x), F_12(x)*F_23(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_10(x) + F_24(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_58(x) + F_60(x))
Eq(F_31(x), F_32(x) + F_9(x))
Eq(F_32(x), F_33(x) + F_42(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_9(x))
Eq(F_36(x), F_33(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_10(x))
Eq(F_41(x), F_37(x))
Eq(F_42(x), F_19(x) + F_22(x) + F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_52(x))
Eq(F_45(x), F_10(x) + F_46(x))
Eq(F_46(x), F_19(x) + F_47(x) + F_50(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_10(x) + F_49(x))
Eq(F_49(x), F_47(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_10(x) + F_46(x))
Eq(F_52(x), F_42(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_46(x))
Eq(F_57(x), F_53(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_12(x)*F_16(x)*F_9(x)**2)
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_30(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_12(x)*F_64(x)*F_65(x)*F_9(x))
Eq(F_64(x), F_27(x) + F_9(x))
Eq(F_65(x), F_66(x) + F_67(x))
Eq(F_66(x), F_9(x)**2)
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_12(x)*F_51(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "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_and_col_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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], 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