01324_01342_03124_03142_30124_30142
Counting sequence:
1, 1, 2, 6, 24, 114, 596, 3298, 18944, 111778, 673220, 4121434, 25570144, 160415810, 1015899124, 6485782866, 41698225920, 269735407170, 1754332330244, 11465139694762, 75252567550880, 495851249128066, 3278760949419764, 21749946173057026, 144702650959854464, 965293531956318946, 6455267260210061252, 43267215303773301498, 290616713169640562400, 1955842939532896306242, 13186837427427729377140, 89061083165066292105266, 602460214294988929154560, 4081489885631942814133378, 27689799482394888480557060, 188103058780867095947509066, 1279422981542485152102130464, 8712537855174324458179510914, 59396365577313030020250046836, 405353865950223976265306597602, 2769137749970459872146687722624, 18935155951953525435331574989858, 129594679893784665321234844581316, 887729656841841390920561791180442, 6086002979537358565684736717352544, 41756612410330305422127709857802818, 286711352214032906274237870325256436, 1970048432205973994288160175106797842, 13545905820241381771524261324738157824, 93202145873849418994659446959437136578, 641680464123440535050565034146461781252, 4420544858704239651499333681057246054890, 30470988366870261681547342633881963697824, 210155763639575882582069602651517618518146, 1450210057895197163032562016833567785986548, 10012595969233063584443159124841622791243714, 69164054041360184493311740031171751476090240, 477996150950420152545811660237664391216173410, 3304998074771315154176041052211335695244378564, 22861983129995642615032150970590027836967770682, 158214340405283153118150996822815148710629757408, 1095369778433246271670572284476746508498351106626, 7586695356639126880179504880949533336407724621940, 52567318456841332350484923142132833155237810605042, 364371890007673543916695083008163914540654550367232, 2526588997930550584777966959273964470088093995907330, 17525890645272216710232261423694119191102435338872836, 121612060690424435218986105382568143889290018558503562, 844150248601158882028807422709158193643492993488074272, 5861450650905907547099701052882418666619391208175475842, 40712579030139625963940907187042108925432010157799209076, 282869628586154616847705155417553206381969938975974177954, 1965959007651756309830473984116295321473615447920725384832, 13667510411326515692512332241949828562677612548227184050850, 95044659585013244999971198098541117464514670007995364690884, 661128760308019381243461580025082735713723578746345155556826, 4600035800990808186317109315087861482583095780348239911713120, 32014751587183300184578494528012099548229690338568671978220610, 222869108354978009763247568402483063191367725363142777884161524, 1551878062812050733622625014240607778695164600280950772242634962, 10808627846524961754331066942684190544326962471113343162139200256, 75298471281955851424758655208161425064537427996000096297671743298, 524688869454857198133822579085506986889074654064256060124671408388, 3656917846055062044689328214088332785752779660365905788114570458922, 25493171537722910926338701972263260235716852228178500001173445357984, 177756536508382444752771748482544965857274420706406776311219987928706, 1239704430000057548684482645244234360766650962541406926628865376296180, 8647677393434997789997536636677901635799140340473665478866191277129602, 60334750737129294323085370021958479361979065211098899832021304722886528, 421037051523473386207222567924353841328367537582103725176856231230152674, 2938704765869125432543044244779551068683689276323819076930040785436867524, 20515051426301301148178774307974728261127950339595268723247382164624869754, 143241380067428793787482950024624829439280321349658364472819751479898051808, 1000326785203634791687643428076696635852909470746137731254994528907855484994, 6987006539187477328337598265028354790854126093722999830097517117460814270836, 48810655644086877788096715178734170897529874234545872967135272914283892559794, 341044313394054553433606768686920889837230452937349789668286303226540844874240, 2383296919602969628063478020065893651164536164957139168524004975831159156385154, 16657706052690300696952710271736931882862230307772599470856055957957510210887684, 116444912732089937096679992313352173035707782299420628109164473914508248858986442, 814128112348200011756490585504765286479707604273092790802853659252653429281601824
Implicit equation for the generating function in Maple syntax:
(2*x-1)*F(x)^3+(-3*x+1)*F(x)^2+2*F(x)-2 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) F \! \left(x \right)^{3}+\left(-3 x +1\right) F \! \left(x \right)^{2}+2 F \! \left(x \right)-2 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(n+5) = 81*n*(2*n+1)/(n+5)/(n+4)*a(n)-9/2*(72*n^2+153*n+80)/(n+5)/(n+4)*a(n+1)+3/2*(n+2)*(167*n+300)/(n+5)/(n+4)*a(n+2)-1/2*(185*n^2+1021*n+1400)/(n+5)/(n+4)*a(n+3)+1/2*(32*n+105)/(n+5)*a(n+4), n >= 5
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) = 24
a \! \left(n +5\right) = \frac{81 n \left(2 n +1\right) a \! \left(n \right)}{\left(n +5\right) \left(n +4\right)}-\frac{9 \left(72 n^{2}+153 n +80\right) a \! \left(n +1\right)}{2 \left(n +5\right) \left(n +4\right)}+\frac{3 \left(n +2\right) \left(167 n +300\right) a \! \left(n +2\right)}{2 \left(n +5\right) \left(n +4\right)}-\frac{\left(185 n^{2}+1021 n +1400\right) a \! \left(n +3\right)}{2 \left(n +5\right) \left(n +4\right)}+\frac{\left(32 n +105\right) a \! \left(n +4\right)}{2 n +10}, \quad n \geq 5
Specification 1
Strategy pack name: row_and_col_placements_tracked_fusion
Tree: http://permpal.com/tree/24093/
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[4,x,1]
F[4,x,k[0]] = F[0,x]+F[5,x,k[0]]
F[5,x,k[0]] = F[12,x,k[0]]*F[6,x,k[0]]
F[6,x,k[0]] = F[10,x,k[0]]+F[4,x,k[0]]+F[7,x,k[0]]
F[7,x,k[0]] = F[8,x,k[0]]*F[9,x]
F[8,x,k[0]] = 1/(-1+k[0])*(F[6,x,k[0]]*k[0]-F[6,x,1])
F[9,x] = x
F[10,x,k[0]] = F[11,x,k[0]]
F[11,x,k[0]] = F[12,x,k[0]]*F[13,x,k[0]]*F[6,x,k[0]]
F[12,x,k[0]] = k[0]*x
F[13,x,k[0]] = F[14,x,k[0]]+F[17,x,k[0]]
F[14,x,k[0]] = F[1,x]+F[15,x,k[0]]
F[15,x,k[0]] = F[16,x,k[0]]
F[16,x,k[0]] = F[12,x,k[0]]*F[14,x,k[0]]
F[17,x,k[0]] = F[18,x,k[0]]+F[21,x,k[0]]
F[18,x,k[0]] = F[19,x,k[0]]
F[19,x,k[0]] = F[12,x,k[0]]*F[20,x,k[0]]
F[20,x,k[0]] = F[1,x]+F[18,x,k[0]]
F[21,x,k[0]] = F[22,x]+F[23,x,k[0]]+F[25,x,k[0]]
F[22,x] = 0
F[23,x,k[0]] = F[12,x,k[0]]*F[24,x,k[0]]
F[24,x,k[0]] = F[15,x,k[0]]+F[21,x,k[0]]
F[25,x,k[0]] = F[12,x,k[0]]*F[17,x,k[0]]
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_{4}\! \left(x , 1\right)
F_{4}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{5}\! \left(x , y\right)
F_{5}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{6}\! \left(x , y\right)
F_{6}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{4}\! \left(x , y\right)+F_{7}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{8}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{8}\! \left(x , y\right) = \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}
F_{9}\! \left(x \right) = x
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{13}\! \left(x , y\right) F_{6}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = y x
F_{13}\! \left(x , y\right) = F_{14}\! \left(x , y\right)+F_{17}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)
F_{15}\! \left(x , y\right) = F_{16}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{14}\! \left(x , y\right)
F_{17}\! \left(x , y\right) = F_{18}\! \left(x , y\right)+F_{21}\! \left(x , y\right)
F_{18}\! \left(x , y\right) = F_{19}\! \left(x , y\right)
F_{19}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{20}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{18}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{22}\! \left(x \right)+F_{23}\! \left(x , y\right)+F_{25}\! \left(x , y\right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{24}\! \left(x , y\right)
F_{24}\! \left(x , y\right) = F_{15}\! \left(x , y\right)+F_{21}\! \left(x , y\right)
F_{25}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{17}\! \left(x , y\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_4(x, 1))
Eq(F_4(x, k_0), F_0(x) + F_5(x, k_0))
Eq(F_5(x, k_0), F_12(x, k_0)*F_6(x, k_0))
Eq(F_6(x, k_0), F_10(x, k_0) + F_4(x, k_0) + F_7(x, k_0))
Eq(F_7(x, k_0), F_8(x, k_0)*F_9(x))
Eq(F_8(x, k_0), (k_0*F_6(x, k_0) - F_6(x, 1))/(k_0 - 1))
Eq(F_9(x), x)
Eq(F_10(x, k_0), F_11(x, k_0))
Eq(F_11(x, k_0), F_12(x, k_0)*F_13(x, k_0)*F_6(x, k_0))
Eq(F_12(x, k_0), k_0*x)
Eq(F_13(x, k_0), F_14(x, k_0) + F_17(x, k_0))
Eq(F_14(x, k_0), F_1(x) + F_15(x, k_0))
Eq(F_15(x, k_0), F_16(x, k_0))
Eq(F_16(x, k_0), F_12(x, k_0)*F_14(x, k_0))
Eq(F_17(x, k_0), F_18(x, k_0) + F_21(x, k_0))
Eq(F_18(x, k_0), F_19(x, k_0))
Eq(F_19(x, k_0), F_12(x, k_0)*F_20(x, k_0))
Eq(F_20(x, k_0), F_1(x) + F_18(x, k_0))
Eq(F_21(x, k_0), F_22(x) + F_23(x, k_0) + F_25(x, k_0))
Eq(F_22(x), 0)
Eq(F_23(x, k_0), F_12(x, k_0)*F_24(x, k_0))
Eq(F_24(x, k_0), F_15(x, k_0) + F_21(x, k_0))
Eq(F_25(x, k_0), F_12(x, k_0)*F_17(x, k_0))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "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_and_col_placements_tracked_fusion", "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, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 4, 2], "pos": [[0, 0], [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, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2, 4], "pos": [[0, 1], [0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2, 4], "pos": [[0, 1], [0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 3, 2, 4], "pos": [[0, 1], [0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2, 4], "pos": [[0, 1], [0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 1], 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