0132_1230_1302_2130_3021
Counting sequence:
1, 1, 2, 6, 19, 53, 136, 334, 805, 1929, 4622, 11094, 26679, 64253, 154900, 373662, 901705, 2176401, 5253658, 12682662, 30617691, 73916485, 178448800, 430811886, 1040069997, 2510948889, 6061964326, 14634873590, 35331707007, 85298282509, 205928266284, 497154808638, 1200237876369, 2897630553377, 6995498974258, 16888628492102, 40772755947683, 98434140375637, 237641036686008, 573716213733518, 1385073464137653, 3343863141992105, 8072799748103742, 19489462638179990, 47051725024442567, 113592912687042333, 274237550398502724, 662068013484021470, 1598373577366517465, 3858815168217026225, 9316003913800537674, 22490822995818067174, 54297649905436635371, 131086122806691298917, 316469895518819191760, 764025913844329638446, 1844521723207478422013, 4453069360259286433081, 10750660443726051235926, 25954390247711388849718, 62659440939148828877071, 151273272126009046542381, 365205985191166921897052, 881685242508342890268286, 2128576470207852702361889, 5138838182924048294916673, 12406252836055949292116066, 29951343855035946879065734, 72308940546127843050160435, 174569224947291632979295349, 421447390440711109008655592, 1017464005828713850996506574, 2456375402098138811001564229, 5930214810024991472999525833, 14316805022148121757000501870, 34563824854321234987000410582, 83444454730790591731001198935, 201452734315902418449002679101, 486349923362595428629006422388, 1174152581041093275707015383582, 2834655085444781980043037043561, 6843462751930657235793089318865, 16521580589306096451629215523450, 39886623930542850139051520201766, 96294828450391796729732255756667, 232476280831326443598516031538309, 561247390113044683926764318649856, 1354971061057415811452044668647790, 3271189512227876306830853655748237, 7897350085513168425113751979939929, 19065889683254213157058357615416454, 46029129452021594739230467210553718, 111124148587297402635519292036297119, 268277426626616400010269051282913357, 647679001840530202656057394601881228, 1563635430307676805322383840486425022, 3774949862455883813300825075574472113, 9113535155219444431924033991635101537, 22002020172894772677148893058844398738, 53117575501008989786221820109323613638, 128237171174912752249592533277491331523
Generating function in Maple syntax:
(2*x^5+5*x^3-9*x^2+5*x-1)/(x^2+2*x-1)/(x-1)^4
Generating function in latex syntax:
\frac{2 x^{5}+5 x^{3}-9 x^{2}+5 x -1}{\left(x^{2}+2 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(2*x**5 + 5*x**3 - 9*x**2 + 5*x - 1)/((x - 1)**4*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+2*x-1)*(x-1)^4*F(x)-2*x^5-5*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)-2 x^{5}-5 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
1/6*n^3-n^2+1/3*n+1/2*(-1-2^(1/2))^(-n)+1/2*(2^(1/2)-1)^(-n)-1/8*(-1-2^(1/2))^(-n)*2^(1/2)+1/8*(2^(1/2)-1)^(-n)*2^(1/2)
Explicit closed form in latex syntax:
\frac{n^{3}}{6}-n^{2}+\frac{n}{3}+\frac{\left(-1-\sqrt{2}\right)^{-n}}{2}+\frac{\left(\sqrt{2}-1\right)^{-n}}{2}-\frac{\left(-1-\sqrt{2}\right)^{-n} \sqrt{2}}{8}+\frac{\left(\sqrt{2}-1\right)^{-n} \sqrt{2}}{8}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(n+2) = -1/3*n^3+2*n^2+a(n)+2*a(n+1)+1/3*n-1, 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) = 53
a \! \left(n +2\right) = -\frac{n^{3}}{3}+2 n^{2}+a \! \left(n \right)+2 a \! \left(n +1\right)+\frac{n}{3}-1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/20059/
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[12,x]*F[4,x]
F[4,x] = F[32,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,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[21,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[14,x]
F[21,x] = F[22,x]+F[23,x]+F[31,x]
F[22,x] = 0
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[10,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[10,x]*F[12,x]
F[28,x] = F[21,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[21,x]
F[31,x] = F[12,x]*F[13,x]
F[32,x] = F[33,x]+F[43,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[1,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[36,x]
F[39,x] = F[33,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[33,x]
F[43,x] = F[22,x]+F[44,x]+F[62,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[51,x]
F[46,x] = F[47,x]+F[6,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[10,x]+F[50,x]
F[50,x] = F[48,x]
F[51,x] = F[43,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[22,x]+F[56,x]+F[60,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[49,x]+F[58,x]
F[58,x] = F[55,x]+F[59,x]
F[59,x] = F[53,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[33,x]+F[55,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[61,x]+F[64,x]
F[64,x] = F[65,x]+F[68,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[37,x]
F[68,x] = 2*F[22,x]+F[69,x]+F[71,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[50,x]
F[71,x] = F[12,x]*F[64,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{32}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \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_{21}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{31}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{28}\! \left(x \right) = F_{21}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{43}\! \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_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \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_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{33}\! \left(x \right)
F_{43}\! \left(x \right) = F_{22}\! \left(x \right)+F_{44}\! \left(x \right)+F_{62}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{51}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{6}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{43}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{22}\! \left(x \right)+F_{56}\! \left(x \right)+F_{60}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{49}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{55}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{53}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{33}\! \left(x \right)+F_{55}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{61}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{37}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{69}\! \left(x \right)+F_{71}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{50}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_32(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_12(x)*F_8(x))
Eq(F_8(x), F_13(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) + F_21(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_14(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_31(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_10(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_10(x)*F_12(x))
Eq(F_28(x), F_21(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_21(x))
Eq(F_31(x), F_12(x)*F_13(x))
Eq(F_32(x), F_33(x) + F_43(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_39(x))
Eq(F_36(x), F_1(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_36(x))
Eq(F_39(x), F_33(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_33(x))
Eq(F_43(x), F_22(x) + F_44(x) + F_62(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_51(x))
Eq(F_46(x), F_47(x) + F_6(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_10(x) + F_50(x))
Eq(F_50(x), F_48(x))
Eq(F_51(x), F_43(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_22(x) + F_56(x) + F_60(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_49(x) + F_58(x))
Eq(F_58(x), F_55(x) + F_59(x))
Eq(F_59(x), F_53(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_33(x) + F_55(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_61(x) + F_64(x))
Eq(F_64(x), F_65(x) + F_68(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_37(x))
Eq(F_68(x), 2*F_22(x) + F_69(x) + F_71(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_50(x))
Eq(F_71(x), F_12(x)*F_64(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, 2, 3, 0], "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, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "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, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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