0132_1203_2013
Counting sequence:
1, 1, 2, 6, 21, 77, 290, 1118, 4398, 17595, 71385, 293042, 1215035, 5081259, 21408350, 90786332, 387212538, 1659936246, 7148446212, 30910820373, 134158126707, 584229340360, 2552004072617, 11178893265681, 49094978794458, 216126986548919, 953536154625977, 4215533308525144, 18672102793409359, 82852127149021417, 368242268545384374, 1639221754619756690, 7307597197794341380, 32621708828448151036, 145814151135541253368, 652562380657623167352, 2923781348821749867934, 13114195452891265438490, 58882709171366528069136, 264643120177116794945341, 1190526650404216735691593, 5360481868398304088461864, 24156644918060198984158051, 108948335095680260308411853, 491745093871962749727482548, 2221161927500496471661152603, 10039853849494889838686758999, 45411793080825060588571149596, 205537892389983185624294543425, 930862998457734026556579894623, 4218320376885759158009111861454, 19126839112690722321436198190409, 86773516885668461065606063587787, 393878773411272461600053630039234, 1788796188121520427061957550662643, 8127819168417158233450774453690511, 36948303192920640769392325633755968, 168040923839214998972343207057077949, 764590482886702569320294240256663877, 3480402681770634070184082213527152106, 15849322969281903656659645703918184497, 72204902335308684374530612930301746151, 329072673248846984569424098165866538262, 1500309287197988494361932232988184817780, 6842716488444158896147535115651031408278, 31219807717101503051505919616793890397888, 142488956683567651781179441229838494116006, 650544599447292995874141535789448067846668, 2971074943593057859397530317937671744294988, 13573340880512524013728847367887708753292760, 62028696186459551355806925528191655376643072, 283548652159802760813258834617886800586734916, 1296546222416812311977077501191171518901072056, 5930215731492460477407264077342604560527284514, 27131373419127652284686843788026568939027757634, 124162005551007358182841379501355712924130523888, 568353177859088770826021587562641902319461982270, 2602301402727559005704190011539310354579313904346, 11918012472483664673366605223155789248362858720936, 54595182766977754116562384838098651428826478195493, 250153436845143552274299581859735057757199310453041, 1146457058419738172764367752420408648327415018714098, 5255401221137924693699360364882192727843781308410877, 24096191927464941387257465264318381310021016190289611, 110505315045868270203920770197437784754735793111702472, 506883299245564249862473911380264232772287821695513485, 2325523748400775094296361633052442137415367499848796881, 10671354500827091565856734706196708808415823634353696774, 48978146543198669652734819136009451971371052135682170413, 224836750246988381153604610087987150839208683880439471933, 1032315843149175622778945382535425445672143366892863957168, 4740634480381131685702401546212968596276300109321551726899, 21773950771472982407613040537799303687637811761958824349511, 100026079318399045612954081646151496761780572489982396404980, 459581891656693926155029572681141296308920437768801578614105, 2111955103502674071778362210434020261485980022826857819505213, 9706822149630584477276287378633674782230852221172776138346660, 44620931605639259648389752106026872888868566571644837542565251, 205148318062007502965727965269235259820057041825034051932729007, 943330102255535840365004852807121538552614553566216424383599078, 4338349243378388425047252880700942647284746759933834321433590159
Generating function in Maple syntax:
1/2*(2*x^2-2*x+1-(-4*x^5+16*x^4-24*x^3+20*x^2-8*x+1)^(1/2))/(x^2-x+1)/x
Generating function in latex syntax:
\frac{2 x^{2}-2 x +1-\sqrt{-4 x^{5}+16 x^{4}-24 x^{3}+20 x^{2}-8 x +1}}{2 \left(x^{2}-x +1\right) x}
Generating function in sympy syntax:
(x**2 - x - sqrt(-4*x**5 + 16*x**4 - 24*x**3 + 20*x**2 - 8*x + 1)/2 + 1/2)/(x*(x**2 - x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-x+1)*x*F(x)^2+(-2*x^2+2*x-1)*F(x)+(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-x +1\right) x F \! \left(x \right)^{2}+\left(-2 x^{2}+2 x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 77
a(6) = 290
a(n+7) = 2*(1+2*n)/(n+8)*a(n)-2*(17+10*n)/(n+8)*a(1+n)+2*(59+22*n)/(n+8)*a(n+2)-12*(18+5*n)/(n+8)*a(n+3)+4*(59+13*n)/(n+8)*a(n+4)-(160+29*n)/(n+8)*a(n+5)+3*(20+3*n)/(n+8)*a(n+6), n >= 7
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) = 21
a \! \left(5\right) = 77
a \! \left(6\right) = 290
a \! \left(n +7\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{2 \left(17+10 n \right) a \! \left(1+n \right)}{n +8}+\frac{2 \left(59+22 n \right) a \! \left(n +2\right)}{n +8}-\frac{12 \left(18+5 n \right) a \! \left(n +3\right)}{n +8}+\frac{4 \left(59+13 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(160+29 n \right) a \! \left(n +5\right)}{n +8}+\frac{3 \left(20+3 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 7
Specification 1
Strategy pack name: row_and_col_placements_tracked_fusion
Tree: http://permpal.com/tree/288/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[17,x]*F[3,x]
F[3,x] = F[0,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[14,x]*F[17,x]*F[6,x]
F[6,x] = F[7,x,1]
F[7,x,k[0]] = F[11,x,k[0]]+F[8,x,k[0]]
F[8,x,k[0]] = F[0,x]+F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]*F[7,x,k[0]]
F[10,x,k[0]] = k[0]*x
F[11,x,k[0]] = F[12,x,k[0]]
F[12,x,k[0]] = F[13,x,k[0]]*F[14,x]*F[17,x]
F[13,x,k[0]] = (F[7,x,k[0]]*k[0]-F[7,x,1])/(-1+k[0])
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[18,x]
F[17,x] = x
F[18,x] = F[19,x]+F[22,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[17,x]*F[19,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[17,x]*F[25,x]
F[25,x] = F[1,x]+F[23,x]
F[26,x] = F[27,x]
F[27,x] = F[17,x]*F[28,x]
F[28,x] = F[20,x]+F[26,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_{17}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{0}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x , 1\right)
F_{7}\! \left(x , y\right) = F_{11}\! \left(x , y\right)+F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{7}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = y x
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{14}\! \left(x \right) F_{17}\! \left(x \right)
F_{13}\! \left(x , y\right) = \frac{y F_{7}\! \left(x , y\right)-F_{7}\! \left(x , 1\right)}{-1+y}
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = x
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{17}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{17}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{20}\! \left(x \right)+F_{26}\! \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_17(x)*F_3(x))
Eq(F_3(x), F_0(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_14(x)*F_17(x)*F_6(x))
Eq(F_6(x), F_7(x, 1))
Eq(F_7(x, k_0), F_11(x, k_0) + F_8(x, k_0))
Eq(F_8(x, k_0), F_0(x) + F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0)*F_7(x, k_0))
Eq(F_10(x, k_0), k_0*x)
Eq(F_11(x, k_0), F_12(x, k_0))
Eq(F_12(x, k_0), F_13(x, k_0)*F_14(x)*F_17(x))
Eq(F_13(x, k_0), (-k_0*F_7(x, k_0) + F_7(x, 1))/(1 - k_0))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_18(x))
Eq(F_17(x), x)
Eq(F_18(x), F_19(x) + F_22(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_17(x)*F_19(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_17(x)*F_25(x))
Eq(F_25(x), F_1(x) + F_23(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_17(x)*F_28(x))
Eq(F_28(x), F_20(x) + F_26(x))
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": 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": [1, 2, 0, 3], "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, 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, 1, 2], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 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": [1, 2, 0, 3], "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": 3, "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], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 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, 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, 1, 2], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", 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