0231_0312_1320_3021
Counting sequence:
1, 1, 2, 6, 20, 66, 219, 734, 2489, 8542, 29655, 104056, 368660, 1317428, 4744046, 17199394, 62731683, 230028726, 847521087, 3136022304, 11648891688, 43421880396, 162373057002, 608949177564, 2289839975826, 8631653808876, 32611149663094, 123466432747664, 468357666961784, 1779894463378664, 6775580302101820, 25833732557249034, 98644629893295895, 377195454806383142, 1444211107644335567, 5536479320056154256, 21249352152008413824, 81646836055922378204, 314043258329377655634, 1209129353218777343268, 4659804462239295047390, 17974395502618884120900, 69392737327140125508210, 268120247730481022918480, 1036776117036975459762040, 4012038732585538747871960, 15536599336242262647404740, 60206638456948244957032340, 233462979889423581291247390, 905868835153149878464657724, 3517024330162065151746234790, 13662744101173363447422438656, 53105969302012335314277789040, 206529679785511202803036589048, 803612557910076745534038618724, 3128440668755361370266239551544, 12184799417668064757815169516580, 47480008860907057049805057798296, 185096502488483028511472445462028, 721895056427903085622780283887520, 2816644939845180787937130767554096, 10994269424423748938909344539892688, 42931021466334290347910390937695224, 167703162216750237486795193510255706, 655347014835950004293841545655467647, 2561866765455245635339199894669964806, 10018258024586923695049735200192367087, 39189874486462974639070985930454186224, 153354753603335478552580811423967995248, 600286023860774827249512027373510742396, 2350460325030911406610719275793646617442, 9206139886272014816449361281200095302484, 36068540648083082078198686876769661788518, 141352170035486413913496678742145017194836, 554110000571857486181301689690666305789402, 2172730787568820145877788839636164381819344, 8521758409699638075384827306692214568671608, 33432045808374386689345503865119930614329976, 131191078720921254056500038182894484793921012, 514932796118693087784868117918103812767473084, 2021617972205609023656174424686470906220134138, 7938663560613239909948924903275499759869628116, 31181216139588368520369473954890470555633071202, 122499329144577219519697464861556891038922306624, 481356814785265575709173118461445123660553266928, 1891869247499280399471718481151253282586323587176, 7437099470711811518872818307039944968398960547372, 29241693099855795749285163174127246415256223402504, 114996847047945609559220304831117716414618171385468, 452326417517965618642517289538884499434087342623336, 1779503081122828074062128296189699741229090665932212, 7002039202605656116731371004267218655829715521472224, 27556726242179777981012880014126301602628604450424848, 108469175510549304582074358767083425713037535526546096, 427030687259118324244341960377292088952589975428056392, 1681451353775880759259370232672517397564190718827484484, 6621867694048023676472746312656613225265772425121734518, 26082319019798359720034023129264414592090663609107545436, 102749567604940894798339405194542810959914259866221830022, 404837308366998805268977014374942360344943643410920904224, 1595315002438446146604588745982384461181108335092041367488
Generating function in Maple syntax:
((x^3+2*x^2-5*x+2)*(1-4*x)^(1/2)+2*x^4+x^3-6*x^2+7*x-2)/(4*x^2-2*x)
Generating function in latex syntax:
\frac{\left(x^{3}+2 x^{2}-5 x +2\right) \sqrt{1-4 x}+2 x^{4}+x^{3}-6 x^{2}+7 x -2}{4 x^{2}-2 x}
Generating function in sympy syntax:
(2*x**4 + x**3 - 6*x**2 + 7*x + sqrt(1 - 4*x)*(x**3 + 2*x**2 - 5*x + 2) - 2)/(4*x**2 - 2*x)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*x*F(x)^2-(2*x-1)*(2*x^2-3*x+2)*(x^2+2*x-1)*F(x)+x^7+2*x^6-2*x^5-3*x^4-4*x^3+15*x^2-10*x+2 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} x F \! \left(x \right)^{2}-\left(2 x -1\right) \left(2 x^{2}-3 x +2\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+x^{7}+2 x^{6}-2 x^{5}-3 x^{4}-4 x^{3}+15 x^{2}-10 x +2 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(6) = 219
a(7) = 734
a(n+5) = -2*(-3+2*n)/(n+6)*a(n)-(7+5*n)/(n+6)*a(1+n)+1/2*(112+51*n)/(n+6)*a(n+2)-(83+24*n)/(n+6)*a(n+3)+1/2*(80+17*n)/(n+6)*a(n+4), n >= 8
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) = 20
a \! \left(5\right) = 66
a \! \left(6\right) = 219
a \! \left(7\right) = 734
a \! \left(n +5\right) = -\frac{2 \left(-3+2 n \right) a \! \left(n \right)}{n +6}-\frac{\left(7+5 n \right) a \! \left(1+n \right)}{n +6}+\frac{\left(112+51 n \right) a \! \left(n +2\right)}{12+2 n}-\frac{\left(83+24 n \right) a \! \left(n +3\right)}{n +6}+\frac{\left(80+17 n \right) a \! \left(n +4\right)}{12+2 n}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21418/
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[13,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[27,x]+F[8,x]
F[8,x] = F[14,x]+F[9,x]
F[9,x] = F[10,x]^3
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]^2*F[13,x]
F[13,x] = x
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[13,x]*F[18,x]*F[25,x]
F[18,x] = F[19,x]+F[24,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[2,x]*F[21,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[13,x]*F[21,x]
F[23,x] = F[11,x]^2
F[24,x] = F[11,x]^2*F[10,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[13,x]*F[25,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[10,x]^2*F[11,x]
F[29,x] = F[2,x]*F[30,x]
F[30,x] = F[22,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{27}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{14}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10} \left(x \right)^{3}
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)^{2} F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right) F_{25}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11} \left(x \right)^{2}
F_{24}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{10}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{25}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{11}\! \left(x \right)
F_{29}\! \left(x \right) = F_{2}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{22}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_13(x)*F_7(x))
Eq(F_7(x), F_27(x) + F_8(x))
Eq(F_8(x), F_14(x) + F_9(x))
Eq(F_9(x), F_10(x)**3)
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)**2*F_13(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_13(x)*F_18(x)*F_25(x))
Eq(F_18(x), F_19(x) + F_24(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_2(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_13(x)*F_21(x))
Eq(F_23(x), F_11(x)**2)
Eq(F_24(x), F_10(x)*F_11(x)**2)
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_13(x)*F_25(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_10(x)**2*F_11(x))
Eq(F_29(x), F_2(x)*F_30(x))
Eq(F_30(x), F_22(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": 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 2], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 2], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, 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