0213_0231_1320_3021
Counting sequence:
1, 1, 2, 6, 20, 66, 219, 738, 2529, 8796, 30975, 110210, 395553, 1430256, 5204968, 19049034, 70064320, 258854666, 960181152, 3574543678, 13351005464, 50015909970, 187885820386, 707579978126, 2670963909402, 10104066863646, 38299489492002, 145444801328454, 553296308825154, 2108253664791134, 8045429646353583, 30746572397505666, 117659940713003041, 450828647638824900, 1729476475916744651, 6642168571429421386, 25537080509791397253, 98282174356506082496, 378615392179273005269, 1459893721553902981230, 5634100861341534137399, 21761592038795079570888, 84120696897897572797737, 325421163719083407906826, 1259809999011700674312755, 4880531790089946677554044, 18919926096728218377531103, 73392224708847709443221222, 284870738937074960556451413, 1106374882050312665177347852, 4299357490276631600722505435, 16716381124521145045967012154, 65029380313878078914194710961, 253102994016149558694995325032, 985592484508058758151052814923, 3839745592645702314770575773286, 14965988416631367994102234696961, 58357882639329802534174579082892, 227655649495603261848154742958295, 888456264601012984505442258554594, 3468699015229320588961077331991325, 13547670402208489338444907101225112, 52932795259156938276654968485622432, 206891304309960705979826506244898522, 808932115306036587365192404581472336, 3163951310764649819628250065380037674, 12379155650458324910168971481340737112, 48449757455796046926892675848321248350, 189682626715308294773578243664988800288, 742839601429295180856738279789328539426, 2909982836217626289761747104451477030614, 11402765250609685316084882892036548667902, 44694177590457458957567702081071918455990, 175230417637611583852631977247447389111318, 687199313351650114740683120255615635085606, 2695675614893206524828320379346766829597174, 10576972313859446206918609725183240159666862, 41510806438199120910783952754691818467109510, 162953759239940710659784678680063584658586188, 639835673910640794803235135920463262045170366, 2512875124839395605720760573964856673276464584, 9871187021637668272212503521519709316332160042, 38784802139240135913613220401157697365644381620, 152421172814758994374525336102768399261401073838, 599126329089161918567218489422542820876901657776, 2355476858967477716392233691371486987253619264754, 9262423224926744306367036740573209244224827726952, 36429551369716061377815210696299444438928646699110, 143306071595065527007532435254263240813390440575268, 563839023727795416974567783963569672323600188910706, 2218828343393394839897732087940049085113290529372224, 8733104043972516262499384079717650651574725399694542, 34378608304835106385887723113203141077317605130505260, 135357119144629065937298930439536504023174778946613578, 533022498278786158916750682502293791224450113338040746, 2099327172521268952871550553469637779720143023723842814, 8269579544132451699530643348974417877872964792766004682, 32580223286388857972620556685596598280115377189274765950, 128378016459577674274330204641071296427430710120651692058, 505931750947882342332118194432278711448108417524308372326, 1994144429159526740065684611030873617897035820320058172874
Generating function in Maple syntax:
1/2*((-2*x^4+4*x^3-6*x^2+4*x-1)*(1-4*x)^(1/2)-2*x^5+6*x^4-14*x^3+14*x^2-6*x+1)/x^2/(x-1)^2
Generating function in latex syntax:
\frac{\left(-2 x^{4}+4 x^{3}-6 x^{2}+4 x -1\right) \sqrt{1-4 x}-2 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{2 x^{2} \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**5 + 3*x**4 - 7*x**3 + 7*x**2 - 3*x + sqrt(1 - 4*x)*(-2*x**4 + 4*x**3 - 6*x**2 + 4*x - 1)/2 + 1/2)/(x**2*(x - 1)**2)
Implicit equation for the generating function in Maple syntax:
x^2*(x-1)^4*F(x)^2+(2*x^5-6*x^4+14*x^3-14*x^2+6*x-1)*(x-1)^2*F(x)+x^8-2*x^7+6*x^6-12*x^5+23*x^4-29*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
x^{2} \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(2 x^{5}-6 x^{4}+14 x^{3}-14 x^{2}+6 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{8}-2 x^{7}+6 x^{6}-12 x^{5}+23 x^{4}-29 x^{3}+20 x^{2}-7 x +1 = 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) = 738
a(8) = 2529
a(n+6) = -4*(-1+2*n)/(8+n)*a(n)-2*(52+23*n)/(8+n)*a(2+n)+2*(7+13*n)/(8+n)*a(n+1)+2*(93+25*n)/(8+n)*a(n+3)-2*(77+15*n)/(8+n)*a(n+4)+(59+9*n)/(8+n)*a(n+5)+2/(8+n), 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) = 20
a \! \left(5\right) = 66
a \! \left(6\right) = 219
a \! \left(7\right) = 738
a \! \left(8\right) = 2529
a \! \left(n +6\right) = -\frac{4 \left(-1+2 n \right) a \! \left(n \right)}{8+n}-\frac{2 \left(52+23 n \right) a \! \left(2+n \right)}{8+n}+\frac{2 \left(7+13 n \right) a \! \left(n +1\right)}{8+n}+\frac{2 \left(93+25 n \right) a \! \left(n +3\right)}{8+n}-\frac{2 \left(77+15 n \right) a \! \left(n +4\right)}{8+n}+\frac{\left(59+9 n \right) a \! \left(n +5\right)}{8+n}+\frac{2}{8+n}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21349/
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[9,x]
F[4,x] = F[10,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[5,x]^2*F[9,x]
F[9,x] = x
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = F[13,x]+F[19,x]
F[13,x] = F[14,x]*F[5,x]
F[14,x] = F[15,x]+F[5,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[5,x]*F[9,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[5,x]*F[7,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]+F[22,x]
F[21,x] = F[5,x]^2*F[7,x]
F[22,x] = F[23,x]*F[26,x]*F[7,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[9,x]
F[25,x] = F[1,x]+F[24,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[26,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_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{4}\! \left(x \right) = F_{10}\! \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_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5} \left(x \right)^{2} F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
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) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{5}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{5}\! \left(x \right) F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{5}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)
F_{21}\! \left(x \right) = F_{5} \left(x \right)^{2} F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{26}\! \left(x \right) F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{9}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{26}\! \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_4(x)*F_9(x))
Eq(F_4(x), F_10(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_8(x))
Eq(F_8(x), F_5(x)**2*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), F_13(x) + F_19(x))
Eq(F_13(x), F_14(x)*F_5(x))
Eq(F_14(x), F_15(x) + F_5(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_5(x)*F_9(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_5(x)*F_7(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x) + F_22(x))
Eq(F_21(x), F_5(x)**2*F_7(x))
Eq(F_22(x), F_23(x)*F_26(x)*F_7(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_9(x))
Eq(F_25(x), F_1(x) + F_24(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_26(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, 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, 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, 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, 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, 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, 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, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [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": [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]]}], "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, 0], [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": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 2], [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, 1], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": 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