0132_0231_1032_1320_3021
Counting sequence:
1, 1, 2, 6, 19, 58, 177, 553, 1781, 5901, 20017, 69179, 242629, 861039, 3085137, 11143048, 40521915, 148228114, 545015157, 2013110338, 7466142495, 27791858284, 103796106657, 388827899728, 1460610788679, 5500653108886, 20763968187293, 78550029228912, 297751663110403, 1130766745528982, 4301779867346713, 16391940591727209, 62556867387322149, 239079067456388533, 914941179678537681, 3505872965480496431, 13449920235922734973, 51657746057670346011, 198617422806081949889, 764436354394059174453, 2944999310441309786593, 11356077517170864869365, 43828002537356929767921, 169292384122645010490625, 654438786715667480526225, 2531809762612314364216417, 9801873981078029924132673, 37974290241831737829915427, 147217714653456656406968157, 571095135274777260227593471, 2216785519777433341262937561, 8609842749641459513879121367, 33459018899762081648400349941, 130097303955149081853605642995, 506117898018471062805504431361, 1969949415546272092155585453859, 7671328063757220323565211578069, 29887574946270920213304956179255, 116495290260046985337550404177801, 454272985836705167962730118708595, 1772188576332574101937986401096157, 6916420186476212716603252935212359, 27003835551688007229609738030958737, 105471881906750239756563604978394776, 412106872506176426991000166446011115, 1610793797622432926021622315836853106, 6298283500748796145870884486443591301, 24634972203012890706059219684751495754, 96388498029444861554150665804054450623, 377257128412085271470265217426546816324, 1477014710192496572288912616951391900689, 5784469004798880842807757395590637274532, 22660518056663018398692752614054204156431, 88797415543477945189356164526706594955842, 348057993390660237162177981450696275368317, 1364647447776169454373465595851163178893180, 5351848280286431601535412924151058195128699, 20994165287740728124918915291730433034574650, 82376258313834321780245508838503254229759945, 323304230745747667120936146195510231758875918, 1269182105997532402404147830071223704228911707, 4983531037693494875196053865230168810038997088, 19572604040924265761660185808756741755874911693, 76887528245678318651469175809042724253490226406, 302104191255500653823616014990810904787773584095, 1187268537374366978327293107558694934748137609976, 4666920453284000601456882474492149097049087526721, 18348434462590559224572308133340184320339729668214, 72152739535442228093075492355963557470501024208283, 283785311410099718338677170992288214736715391640464, 1116370688648636226206764570951016260453926356687509, 4392447414478469448681814078900198292065875928300482, 17285526697232892271308282215659118086213684172384111, 68035391337511961401976367770187718970541559877844252, 267831564070170983012891757952030470826618942411453417, 1054536596814785033641506594979398749862065435816631880, 4152723929046651064466611790674848098086639864703652639, 16355901804551040176886065501082236396562659565395841790, 64429465446209544968587205228783949453618598782188507925, 253841002455153834731766239869016950579232836049580484768, 1000241541104973637422851665087286751395907030495693448587
Generating function in Maple syntax:
1/4*((-4*x^3+6*x^2-4*x+1)*(1-4*x)^(1/2)-2*x^5-4*x^4+6*x^3-6*x^2+4*x-1)/(x-1/2)/x/(x-1)^2
Generating function in latex syntax:
\frac{\left(-4 x^{3}+6 x^{2}-4 x +1\right) \sqrt{1-4 x}-2 x^{5}-4 x^{4}+6 x^{3}-6 x^{2}+4 x -1}{4 \left(x -\frac{1}{2}\right) x \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**5/2 - x**4 + 3*x**3/2 - 3*x**2/2 + x + sqrt(1 - 4*x)*(-4*x**3 + 6*x**2 - 4*x + 1)/4 - 1/4)/(x*(x - 1)**2*(x - 1/2))
Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x-1)^4*F(x)^2+(2*x-1)*(2*x^5+4*x^4-6*x^3+6*x^2-4*x+1)*(x-1)^2*F(x)+x^9+4*x^8-2*x^7+10*x^6-35*x^5+55*x^4-50*x^3+27*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(2 x -1\right) \left(2 x^{5}+4 x^{4}-6 x^{3}+6 x^{2}-4 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{9}+4 x^{8}-2 x^{7}+10 x^{6}-35 x^{5}+55 x^{4}-50 x^{3}+27 x^{2}-8 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 58
a(6) = 177
a(7) = 553
a(8) = 1781
a(n+6) = -16*(2*n+1)/(n+7)*a(n)+8*(13*n+16)/(n+7)*a(1+n)-4*(36*n+83)/(n+7)*a(n+2)+2*(55*n+192)/(n+7)*a(n+3)-16*(3*n+14)/(n+7)*a(n+4)+(64+11*n)/(n+7)*a(n+5)+3/(n+7), n >= 9
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) = 58
a \! \left(6\right) = 177
a \! \left(7\right) = 553
a \! \left(8\right) = 1781
a \! \left(n +6\right) = -\frac{16 \left(2 n +1\right) a \! \left(n \right)}{n +7}+\frac{8 \left(13 n +16\right) a \! \left(1+n \right)}{n +7}-\frac{4 \left(36 n +83\right) a \! \left(n +2\right)}{n +7}+\frac{2 \left(55 n +192\right) a \! \left(n +3\right)}{n +7}-\frac{16 \left(3 n +14\right) a \! \left(n +4\right)}{n +7}+\frac{\left(64+11 n \right) a \! \left(n +5\right)}{n +7}+\frac{3}{n +7}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21703/
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[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[12,x]*F[7,x]
F[7,x] = F[13,x]+F[8,x]
F[8,x] = F[9,x]^3
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[9,x]^2*F[12,x]
F[12,x] = x
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[15,x]
F[15,x] = F[16,x]+F[27,x]
F[16,x] = F[17,x]+F[23,x]
F[17,x] = F[18,x]*F[20,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[12,x]*F[18,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[18,x]*F[20,x]
F[23,x] = F[24,x]*F[25,x]
F[24,x] = F[10,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[12,x]*F[25,x]
F[27,x] = F[24,x]*F[25,x]*F[9,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_{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_{12}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{13}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9} \left(x \right)^{3}
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_{9} \left(x \right)^{2} F_{12}\! \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_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{27}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{23}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{20}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right) F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = 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_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{27}\! \left(x \right) = F_{24}\! \left(x \right) F_{25}\! \left(x \right) F_{9}\! \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_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_12(x)*F_7(x))
Eq(F_7(x), F_13(x) + F_8(x))
Eq(F_8(x), F_9(x)**3)
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)**2)
Eq(F_12(x), x)
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_15(x))
Eq(F_15(x), F_16(x) + F_27(x))
Eq(F_16(x), F_17(x) + F_23(x))
Eq(F_17(x), F_18(x)*F_20(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_12(x)*F_18(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_18(x)*F_20(x))
Eq(F_23(x), F_24(x)*F_25(x))
Eq(F_24(x), F_10(x))
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_12(x)*F_25(x))
Eq(F_27(x), F_24(x)*F_25(x)*F_9(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, 1, 3, 2], "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, 0, 3, 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, 1, 3, 2], "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, 0, 3, 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, 1, 3, 2], "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, 0, 3, 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"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "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, 0, 3, 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, 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, 0], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], 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