0123_0132_0213_0231_2013
Counting sequence:
1, 1, 2, 6, 19, 62, 209, 722, 2538, 9040, 32539, 118135, 431997, 1589435, 5878927, 21845070, 81502399, 305177172, 1146393763, 4318934289, 16314045437, 61771311906, 234403129145, 891279570779, 3395239193128, 12956045755416, 49518415288124, 189541967503029, 726516612021349, 2788358589170781, 10714692335642685, 41219917915574585, 158745728951661253, 611981318070362365, 2361518578853997813, 9120938486162824110, 35258373670792979228, 136408307803691769681, 528150607243302178893, 2046430770378888268898, 7934940750048544226643, 30788162033950191261594, 119537540574034456909480, 464401100302228085084939, 1805257732804025833943912, 7021528041010809886968548, 27325024239294927128792886, 106393877091420680568280764, 414468075992307221525969395, 1615383448417665915376093572, 6298867333695788224977833831, 24572209779409093207498428044, 95898965575718211468001565273, 374425135458500909081832429185, 1462484224737924685268790425729, 5714609868604901373868049254492, 22338065850095981761983909083395, 87349976475512882166377218366543, 341690722842020673021671150886584, 1337063155963401946291497049731810, 5233767100779320118609501754797164, 20493503713931847671313753669422725, 80269988720275538127277723724492009, 314500513052432682340663105834388313, 1232584963522105945128131633624503602, 4832101768607907457637613009797128994, 18948528339366907404378584825132104214, 74324459582132459951722235377682605211, 291609515235772737236618062687168428997, 1144411479217905938205993612163866644771, 4492316599720468744266343560123999095329, 17638567567349651406575208971682071855981, 69272100920259939493221578860656486528866, 272115251238057096664695983962374162227962, 1069164115865122283225225805915371346144131, 4201752008404885865273572065934793714547021, 16516144915949628440833224239476967329900880, 64934715288698779255536279155877079690991417, 255348332892785714975959290275048367807851648, 1004326365353182005108918783921433776248240142, 3950940110447674876446304943954279925923344187, 15545612581027247612986883855419307141692094930, 61177986245461174640141715759823520841191544828, 240802313455007483405875531468976723532557865694, 947987324925523053721580364012648300140626286986, 3732665913285867353066745344683548932225341945141, 14699711037609080138553390817766421364678386598962, 57898856516237644307938617673015091259751893275203, 228087331614952822795177219720444075113516726054298, 898671267191578820441555296439384250054256971966699, 3541340079954744909804572976356589066088184811901889, 13957258309814424260345714420830488404536005492458377, 55017007015795247590480394085074292331163992450325264, 216898660019220024515596601926235442363159512065466205, 855221443452618381013492953359586303250740872544269813, 3372569256704560615208271630788602394986940699412466326, 13301559730559500273716237500573461975927212816137144150, 52468966235629215120178837492052389463686706760021267064, 206994866828883970420449072106217162689190367820325457745, 816719055195873527820260508340859273260665243314517350403, 3222855187854028146302646705019812015178315987091524817511
Generating function in Maple syntax:
((-x^2+x-1)*(1-4*x)^(1/2)-2*x^4+6*x^3-9*x^2+7*x-1)/(2*x^5-8*x^4+16*x^3-18*x^2+12*x-2)
Generating function in latex syntax:
\frac{\left(-x^{2}+x -1\right) \sqrt{1-4 x}-2 x^{4}+6 x^{3}-9 x^{2}+7 x -1}{2 x^{5}-8 x^{4}+16 x^{3}-18 x^{2}+12 x -2}
Generating function in sympy syntax:
(-2*x**4 + 6*x**3 - 9*x**2 + 7*x + sqrt(1 - 4*x)*(-x**2 + x - 1) - 1)/(2*x**5 - 8*x**4 + 16*x**3 - 18*x**2 + 12*x - 2)
Implicit equation for the generating function in Maple syntax:
(x^5-4*x^4+8*x^3-9*x^2+6*x-1)*F(x)^2+(2*x^4-6*x^3+9*x^2-7*x+1)*F(x)+x*(x^2-2*x+2) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}-4 x^{4}+8 x^{3}-9 x^{2}+6 x -1\right) F \! \left(x \right)^{2}+\left(2 x^{4}-6 x^{3}+9 x^{2}-7 x +1\right) F \! \left(x \right)+x \left(x^{2}-2 x +2\right) = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 62
a(6) = 209
a(7) = 722
a(n+8) = -2*(5+2*n)/(n+8)*a(n)+(62+21*n)/(n+8)*a(n+1)-(200+57*n)/(n+8)*a(n+2)+(402+97*n)/(n+8)*a(n+3)-(546+113*n)/(n+8)*a(n+4)+(494+87*n)/(n+8)*a(n+5)-22*(13+2*n)/(n+8)*a(n+6)+(80+11*n)/(n+8)*a(n+7), 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) = 19
a \! \left(5\right) = 62
a \! \left(6\right) = 209
a \! \left(7\right) = 722
a \! \left(n +8\right) = -\frac{2 \left(5+2 n \right) a \! \left(n \right)}{n +8}+\frac{\left(62+21 n \right) a \! \left(n +1\right)}{n +8}-\frac{\left(200+57 n \right) a \! \left(n +2\right)}{n +8}+\frac{\left(402+97 n \right) a \! \left(n +3\right)}{n +8}-\frac{\left(546+113 n \right) a \! \left(n +4\right)}{n +8}+\frac{\left(494+87 n \right) a \! \left(n +5\right)}{n +8}-\frac{22 \left(13+2 n \right) a \! \left(n +6\right)}{n +8}+\frac{\left(80+11 n \right) a \! \left(n +7\right)}{n +8}, \quad n \geq 8
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21479/
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[9,x]
F[3,x] = F[1,x]+F[25,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[10,x]*F[6,x]*F[9,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[6,x]*F[9,x]
F[9,x] = x
F[10,x] = F[11,x,1]
F[11,x,k[0]] = F[1,x]+F[12,x,k[0]]+F[14,x,k[0]]
F[12,x,k[0]] = F[11,x,k[0]]*F[13,x,k[0]]
F[13,x,k[0]] = k[0]*x
F[14,x,k[0]] = F[15,x,k[0]]*F[9,x]
F[15,x,k[0]] = F[1,x]+F[16,x,k[0]]+F[18,x,k[0]]+F[23,x,k[0]]
F[16,x,k[0]] = F[13,x,k[0]]*F[17,x,k[0]]
F[17,x,k[0]] = (F[11,x,k[0]]*k[0]-F[11,x,1])/(-1+k[0])
F[18,x,k[0]] = F[19,x,k[0]]
F[19,x,k[0]] = F[10,x]*F[20,x,k[0]]*F[6,x]*F[9,x]
F[20,x,k[0]] = F[1,x]+F[21,x,k[0]]
F[21,x,k[0]] = F[22,x,k[0]]
F[22,x,k[0]] = F[13,x,k[0]]*F[20,x,k[0]]
F[23,x,k[0]] = F[24,x,k[0]]*F[9,x]
F[24,x,k[0]] = (F[15,x,k[0]]*k[0]-F[15,x,1])/(-1+k[0])
F[25,x] = F[14,x,1]
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_{9}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{10}\! \left(x \right) F_{6}\! \left(x \right) F_{9}\! \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_{6}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x , 1\right)
F_{11}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{14}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{11}\! \left(x , y\right) F_{13}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = y x
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{15}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{18}\! \left(x , y\right)+F_{23}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{17}\! \left(x , y\right)
F_{17}\! \left(x , y\right) = \frac{y F_{11}\! \left(x , y\right)-F_{11}\! \left(x , 1\right)}{-1+y}
F_{18}\! \left(x , y\right) = F_{19}\! \left(x , y\right)
F_{19}\! \left(x , y\right) = F_{10}\! \left(x \right) F_{20}\! \left(x , y\right) F_{6}\! \left(x \right) F_{9}\! \left(x \right)
F_{20}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{21}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{22}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{20}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = F_{24}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{24}\! \left(x , y\right) = \frac{y F_{15}\! \left(x , y\right)-F_{15}\! \left(x , 1\right)}{-1+y}
F_{25}\! \left(x \right) = F_{14}\! \left(x , 1\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)*F_9(x))
Eq(F_3(x), F_1(x) + F_25(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_10(x)*F_6(x)*F_9(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_6(x)*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x, 1))
Eq(F_11(x, k_0), F_1(x) + F_12(x, k_0) + F_14(x, k_0))
Eq(F_12(x, k_0), F_11(x, k_0)*F_13(x, k_0))
Eq(F_13(x, k_0), k_0*x)
Eq(F_14(x, k_0), F_15(x, k_0)*F_9(x))
Eq(F_15(x, k_0), F_1(x) + F_16(x, k_0) + F_18(x, k_0) + F_23(x, k_0))
Eq(F_16(x, k_0), F_13(x, k_0)*F_17(x, k_0))
Eq(F_17(x, k_0), (-k_0*F_11(x, k_0) + F_11(x, 1))/(1 - k_0))
Eq(F_18(x, k_0), F_19(x, k_0))
Eq(F_19(x, k_0), F_10(x)*F_20(x, k_0)*F_6(x)*F_9(x))
Eq(F_20(x, k_0), F_1(x) + F_21(x, k_0))
Eq(F_21(x, k_0), F_22(x, k_0))
Eq(F_22(x, k_0), F_13(x, k_0)*F_20(x, k_0))
Eq(F_23(x, k_0), F_24(x, k_0)*F_9(x))
Eq(F_24(x, k_0), (-k_0*F_15(x, k_0) + F_15(x, 1))/(1 - k_0))
Eq(F_25(x), F_14(x, 1))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "row_placements_tracked_fusion", "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [2, 0, 1, 3], "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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"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, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 1, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": 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