0132_1032_1203_1230_3102
Counting sequence:
1, 1, 2, 6, 19, 51, 119, 257, 533, 1083, 2180, 4372, 8759, 17548, 35165, 70482, 141276, 283155, 567423, 1136831, 2277113, 4560113, 9130128, 18276738, 36580699, 73206106, 146485329, 293089872, 586373688, 1173062393, 2346635875, 4694100277, 9389542909, 18781259783, 37566039340, 75137776272, 150284774191, 300584472360, 601193095573, 1202425271726, 2404913781188, 4809929887583, 9620025345623, 19240318595083, 38481070673329, 76962842743261, 153926820376664, 307855477051262, 615713925302643, 1231432658116278, 2462873094957545, 4925758776165932, 9851537917323584, 19703108785906693, 39406270888082715, 78812628043713537, 157625395671264981, 315251017193937715, 630502399823144148, 1261005390932989196, 2522011738587973863, 5044025025184671428, 10088052555100090893, 20176109162939683978, 40352224883349650284, 80704460376909118891, 161408937921498373999, 322817903620886739927, 645635852187343647273, 1291271777098147495113, 2582543671865325401440, 5165087534123141459962, 10330175376307804035659, 20660351251069619893778, 41320703308654772777665, 82641407922279090422152, 165282817956043258757896, 330565639328541140355345, 661131284185021981525523, 1322262577314438286769741, 2644525169101210598140013, 5289050361619149544669023, 10578100761127361462400188, 21156201583560513646327048, 42312403266315880387321439, 84624806693132404590918496, 169249613645960306092864709, 338499227712116752913120742, 676998456104125143464751828, 1353996913308338065295500023, 2707993828396655546595604935, 5415987659673378287561912867, 10831975324006803185499239009, 21663950655553720175744703413, 43327901323307600766611158440, 86655802666355475753090410742, 173311605364651386141170787811, 346623210780983481137199760750, 693246421645588105764247802073, 1386492843426478063873202203236, 2772985687071879123580959423888
Generating function in Maple syntax:
(x^10-3*x^9-2*x^8+8*x^7-x^6+2*x^4-13*x^3+14*x^2-6*x+1)/(2*x-1)/(x^2+x-1)/(x-1)^4
Generating function in latex syntax:
\frac{x^{10}-3 x^{9}-2 x^{8}+8 x^{7}-x^{6}+2 x^{4}-13 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**10 - 3*x**9 - 2*x**8 + 8*x**7 - x**6 + 2*x**4 - 13*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**4*(2*x - 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x^2+x-1)*(x-1)^4*F(x)+x^10-3*x^9-2*x^8+8*x^7-x^6+2*x^4-13*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{10}-3 x^{9}-2 x^{8}+8 x^{7}-x^{6}+2 x^{4}-13 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
1/6*n^3-1/2*n^2+1/2*piecewise(n = 0,-67/8,n = 1,-11/4,n = 2,1/2,n = 3,1,0)-11/3*n+4+3/10*(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)-3/10*(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+35/16*2^n-1/2*(-1/2*5^(1/2)-1/2)^(-n)-1/2*(1/2*5^(1/2)-1/2)^(-n)
Explicit closed form in latex syntax:
\frac{n^{3}}{6}-\frac{n^{2}}{2}+\frac{\left(\left\{\begin{array}{cc}-\frac{67}{8} & n =0 \\ -\frac{11}{4} & n =1 \\ \frac{1}{2} & n =2 \\ 1 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)}{2}-\frac{11 n}{3}+4+\frac{3 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{10}-\frac{3 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{10}+\frac{35 \,2^{n}}{16}-\frac{\left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{2}-\frac{\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{2}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 119
a(7) = 257
a(8) = 533
a(9) = 1083
a(10) = 2180
a(n+3) = 1/6*n^3-3/2*n^2-2*a(n)-a(n+1)+3*a(n+2)-8/3*n+13, n >= 11
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) = 51
a \! \left(6\right) = 119
a \! \left(7\right) = 257
a \! \left(8\right) = 533
a \! \left(9\right) = 1083
a \! \left(10\right) = 2180
a \! \left(n +3\right) = \frac{n^{3}}{6}-\frac{3 n^{2}}{2}-2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)-\frac{8 n}{3}+13, \quad n \geq 11
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19654/
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[29,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[21,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[14,x]
F[21,x] = F[19,x]+F[22,x]+F[23,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[25,x]*F[4,x]
F[28,x] = F[21,x]
F[29,x] = F[30,x]+F[40,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[1,x]+F[14,x]
F[34,x] = F[30,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[30,x]
F[39,x] = F[35,x]
F[40,x] = F[22,x]+F[41,x]+F[50,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[11,x]+F[21,x]
F[44,x] = F[45,x]+F[47,x]
F[45,x] = F[22,x]+F[41,x]+F[46,x]
F[46,x] = F[38,x]*F[4,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[35,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[38,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[22,x]+F[54,x]+F[64,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[61,x]
F[56,x] = F[11,x]+F[57,x]
F[57,x] = F[19,x]+F[22,x]+F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[25,x]+F[60,x]
F[60,x] = F[18,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[46,x]
F[63,x] = F[48,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[38,x]+F[66,x]
F[66,x] = F[53,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_{29}\! \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_{21}\! \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_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{21}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{40}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
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_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{30}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{22}\! \left(x \right)+F_{41}\! \left(x \right)+F_{50}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{11}\! \left(x \right)+F_{21}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{22}\! \left(x \right)+F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{35}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{38}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{22}\! \left(x \right)+F_{54}\! \left(x \right)+F_{64}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{61}\! \left(x \right)
F_{56}\! \left(x \right) = F_{11}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{25}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{18}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{46}\! \left(x \right)
F_{63}\! \left(x \right) = F_{48}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{38}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{53}\! \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_29(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_21(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_14(x))
Eq(F_21(x), F_19(x) + F_22(x) + F_23(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_25(x)*F_4(x))
Eq(F_28(x), F_21(x))
Eq(F_29(x), F_30(x) + F_40(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_1(x) + F_14(x))
Eq(F_34(x), F_30(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_30(x))
Eq(F_39(x), F_35(x))
Eq(F_40(x), F_22(x) + F_41(x) + F_50(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_11(x) + F_21(x))
Eq(F_44(x), F_45(x) + F_47(x))
Eq(F_45(x), F_22(x) + F_41(x) + F_46(x))
Eq(F_46(x), F_38(x)*F_4(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_35(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_38(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_22(x) + F_54(x) + F_64(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_61(x))
Eq(F_56(x), F_11(x) + F_57(x))
Eq(F_57(x), F_19(x) + F_22(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_25(x) + F_60(x))
Eq(F_60(x), F_18(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_46(x))
Eq(F_63(x), F_48(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_38(x) + F_66(x))
Eq(F_66(x), F_53(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": [1, 0, 3, 2], "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": [3, 1, 0, 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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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": [3, 1, 0, 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