0123_0132_0213_0231_1230_2301_3012
Counting sequence:
1, 1, 2, 6, 17, 43, 100, 220, 467, 969, 1982, 4018, 8101, 16279, 32648, 65400, 130919, 261973, 524098, 1048366, 2096921, 4194051, 8388332, 16776916, 33554107, 67108513, 134217350, 268435050, 536870477, 1073741359, 2147483152, 4294966768, 8589934031, 17179868589, 34359737738, 68719476070, 137438952769, 274877906203, 549755813108, 1099511626956, 2199023254691, 4398046510201, 8796093021262, 17592186043426, 35184372087797, 70368744176583, 140737488354200, 281474976709480, 562949953420087, 1125899906841349, 2251799813683922, 4503599627369118, 9007199254739561, 18014398509480499, 36028797018962428, 72057594037926340, 144115188075854219, 288230376151710033, 576460752303421718, 1152921504606845146, 2305843009213692061, 4611686018427385951, 9223372036854773792, 18446744073709549536, 36893488147419101087, 73786976294838204253, 147573952589676410650, 295147905179352823510, 590295810358705649297, 1180591620717411300939, 2361183241434822604292, 4722366482869645211068, 9444732965739290424691, 18889465931478580852009, 37778931862957161706718, 75557863725914323416210, 151115727451828646835269, 302231454903657293673463, 604462909807314587349928, 1208925819614629174702936, 2417851639229258349409031, 4835703278458516698821301, 9671406556917033397645922, 19342813113834066795295246, 38685626227668133590593977, 77371252455336267181191523, 154742504910672534362386700, 309485009821345068724777140, 618970019642690137449558107, 1237940039285380274899120129, 2475880078570760549798244262, 4951760157141521099596492618, 9903520314283042199192989421, 19807040628566084398385983119, 39614081257132168796771970608, 79228162514264337593543945680, 158456325028528675187087895919, 316912650057057350374175796493, 633825300114114700748351597738, 1267650600228229401496703200326, 2535301200456458802993406405601
Generating function in Maple syntax:
-(2*x^3-6*x^2+4*x-1)/(2*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{2 x^{3}-6 x^{2}+4 x -1}{\left(2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-2*x**3 + 6*x**2 - 4*x + 1)/((x - 1)**3*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^3*F(x)+2*x^3-6*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+2 x^{3}-6 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
-1/2*n^2-3/2*n-1+2^(n+1)
Explicit closed form in latex syntax:
-\frac{n^{2}}{2}-\frac{3 n}{2}-1+2^{n +1}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+1) = 2*a(n)+1/2*(n+2)*(n-1), n >= 4
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(n +1\right) = 2 a \! \left(n \right)+\frac{\left(n +2\right) \left(n -1\right)}{2}, \quad n \geq 4
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/22717/
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[4,x]
F[3,x] = x
F[4,x] = F[1,x]+F[33,x]+F[5,x]
F[5,x] = F[3,x]*F[6,x]
F[6,x] = F[19,x]+F[7,x]
F[7,x] = F[1,x]+F[5,x]+F[8,x]
F[8,x] = F[3,x]*F[9,x]
F[9,x] = F[10,x]+F[12,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[10,x]*F[3,x]
F[12,x] = F[13,x]+F[14,x]
F[13,x] = F[11,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[3,x]
F[17,x] = F[13,x]+F[14,x]
F[18,x] = F[12,x]*F[3,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[3,x]
F[21,x] = F[20,x]+F[22,x]+F[23,x]
F[22,x] = F[10,x]^2
F[23,x] = F[24,x]
F[24,x] = F[10,x]*F[25,x]*F[3,x]
F[25,x] = F[12,x]+F[26,x]
F[26,x] = F[10,x]*F[27,x]
F[27,x] = F[10,x]+F[28,x]
F[28,x] = F[13,x]+F[29,x]
F[29,x] = F[15,x]+F[30,x]+F[32,x]
F[30,x] = F[3,x]*F[31,x]
F[31,x] = F[13,x]+F[29,x]
F[32,x] = F[28,x]*F[3,x]
F[33,x] = F[3,x]*F[34,x]
F[34,x] = F[35,x]+F[39,x]
F[35,x] = F[10,x]+F[36,x]
F[36,x] = F[3,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[3,x]*F[36,x]
F[39,x] = F[40,x]+F[66,x]
F[40,x] = F[41,x]+F[52,x]
F[41,x] = F[42,x]
F[42,x] = F[3,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[1,x]+F[3,x]
F[45,x] = F[41,x]+F[46,x]
F[46,x] = F[15,x]+F[47,x]+F[51,x]
F[47,x] = F[3,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[3,x]
F[50,x] = F[46,x]
F[51,x] = F[13,x]*F[3,x]
F[52,x] = F[15,x]+F[18,x]+F[53,x]
F[53,x] = F[3,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[13,x]+F[37,x]
F[56,x] = F[52,x]+F[57,x]
F[57,x] = 2*F[15,x]+F[58,x]+F[62,x]
F[58,x] = F[3,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[37,x]
F[61,x] = F[57,x]
F[62,x] = F[3,x]*F[63,x]
F[63,x] = F[64,x]+F[65,x]
F[64,x] = F[51,x]
F[65,x] = F[62,x]
F[66,x] = F[46,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_{4}\! \left(x \right)
F_{3}\! \left(x \right) = x
F_{4}\! \left(x \right) = F_{1}\! \left(x \right)+F_{33}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{3}\! \left(x \right) F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{19}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{5}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{3}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{12}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{10}\! \left(x \right) F_{3}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{3}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{3}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{3}\! \left(x \right)
F_{21}\! \left(x \right) = F_{20}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10} \left(x \right)^{2}
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right) F_{3}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{3}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right) F_{3}\! \left(x \right)
F_{33}\! \left(x \right) = F_{3}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{39}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{3}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{3}\! \left(x \right) F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{66}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{52}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{3}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{1}\! \left(x \right)+F_{3}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{51}\! \left(x \right)
F_{47}\! \left(x \right) = F_{3}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{3}\! \left(x \right)
F_{50}\! \left(x \right) = F_{46}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right) F_{3}\! \left(x \right)
F_{52}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{3}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right)+F_{37}\! \left(x \right)
F_{56}\! \left(x \right) = F_{52}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{3}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{37}\! \left(x \right)
F_{61}\! \left(x \right) = F_{57}\! \left(x \right)
F_{62}\! \left(x \right) = F_{3}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{65}\! \left(x \right)
F_{64}\! \left(x \right) = F_{51}\! \left(x \right)
F_{65}\! \left(x \right) = F_{62}\! \left(x \right)
F_{66}\! \left(x \right) = F_{46}\! \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)*F_4(x))
Eq(F_3(x), x)
Eq(F_4(x), F_1(x) + F_33(x) + F_5(x))
Eq(F_5(x), F_3(x)*F_6(x))
Eq(F_6(x), F_19(x) + F_7(x))
Eq(F_7(x), F_1(x) + F_5(x) + F_8(x))
Eq(F_8(x), F_3(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_12(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_10(x)*F_3(x))
Eq(F_12(x), F_13(x) + F_14(x))
Eq(F_13(x), F_11(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_3(x))
Eq(F_17(x), F_13(x) + F_14(x))
Eq(F_18(x), F_12(x)*F_3(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_3(x))
Eq(F_21(x), F_20(x) + F_22(x) + F_23(x))
Eq(F_22(x), F_10(x)**2)
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_10(x)*F_25(x)*F_3(x))
Eq(F_25(x), F_12(x) + F_26(x))
Eq(F_26(x), F_10(x)*F_27(x))
Eq(F_27(x), F_10(x) + F_28(x))
Eq(F_28(x), F_13(x) + F_29(x))
Eq(F_29(x), F_15(x) + F_30(x) + F_32(x))
Eq(F_30(x), F_3(x)*F_31(x))
Eq(F_31(x), F_13(x) + F_29(x))
Eq(F_32(x), F_28(x)*F_3(x))
Eq(F_33(x), F_3(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_39(x))
Eq(F_35(x), F_10(x) + F_36(x))
Eq(F_36(x), F_3(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_3(x)*F_36(x))
Eq(F_39(x), F_40(x) + F_66(x))
Eq(F_40(x), F_41(x) + F_52(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_3(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_1(x) + F_3(x))
Eq(F_45(x), F_41(x) + F_46(x))
Eq(F_46(x), F_15(x) + F_47(x) + F_51(x))
Eq(F_47(x), F_3(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_3(x))
Eq(F_50(x), F_46(x))
Eq(F_51(x), F_13(x)*F_3(x))
Eq(F_52(x), F_15(x) + F_18(x) + F_53(x))
Eq(F_53(x), F_3(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_13(x) + F_37(x))
Eq(F_56(x), F_52(x) + F_57(x))
Eq(F_57(x), 2*F_15(x) + F_58(x) + F_62(x))
Eq(F_58(x), F_3(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_37(x))
Eq(F_61(x), F_57(x))
Eq(F_62(x), F_3(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_65(x))
Eq(F_64(x), F_51(x))
Eq(F_65(x), F_62(x))
Eq(F_66(x), F_46(x) + F_57(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, 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": [0, 2, 3, 1], "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]]}, {"patt": [3, 0, 1, 2], "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": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 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