0132_0231_1023_2013
Counting sequence:
1, 1, 2, 6, 20, 66, 220, 749, 2596, 9116, 32346, 115795, 417734, 1517112, 5542232, 20351957, 75082156, 278143796, 1034255018, 3858867294, 14442244254, 54204795170, 203971049662, 769375988736, 2908503057030, 11017709699170, 41815905650682, 158987842522785, 605491327278242, 2309551981688976, 8822299387612136, 33746771729222529, 129254156105619844, 495663579076593256, 1902964465394294850, 7313896634905584422, 28139527427575375846, 108370696626065375534, 417746617633980432990, 1611760400998361142256, 6223809808218110769382, 24052682486050436738698, 93026422709057007631890, 360055941569783422376968, 1394571272147931071650194, 5405123249383794256523162, 20962956031420971604343534, 81352348763213121985987268, 315898534218925692621827182, 1227368524599022998370537682, 4771365650563792626205811954, 18558453710351514814541088750, 72221044275061332055042772754, 281190473138132028740544171086, 1095329300361738622332992936170, 4268633512198211431725490181058, 16642794928255949856661073678482, 64915796899845020767321693752690, 253311609774452816780661396210530, 988859296616528290776744332244689, 3861739952541388360700556233278514, 15086737925223873600498527133830836, 58961210204873251387772838365272304, 230510817789720887040955130285822285, 901499498689146735819474835391208684, 3526829823265168897306927556657812932, 13802067000070441231090709468294485754, 54030666728245656900530499101033192602, 211577372584096843449710567982415458894, 828756793512055017065411931484683093274, 3247210305410551254578531420571446632614, 12726698473126439456267599638206710054100, 49893025776952976083702417613405655104702, 195649858065893849907562060823210380400390, 767417711618771026819779070029669517328170, 3010882328733695205492794068864599317695808, 11815786586728630105371777308243930137996442, 46380529008361119473790021427146265842204654, 182100179725368240817674006191636980397330078, 715128648176764541243437821465425400105224508, 2809019164614579367438015907309393582577287582, 11036201901961521170343250799931164117338008070, 43368734345605905148600733181626420280609996586, 170460548104096538869561235505295366931347007066, 670129836420797809390514914563412051884517431530, 2634996007917741439257369714701961613489721470362, 10362987179329757884831478352884542972768688365170, 40763557339805008168689001742885660874402428976902, 160376043831722078206881968139119360294674902511818, 631081522291503902538549597256379002853298217494618, 2483752314712088205214685132095758759120332580668842, 9777015742317409105965428421162181112158287072305128, 38492663453532812327214874717051957720146903182931746, 151572944916381016600708332732609428997701559523894470, 596947310151556324625080471701045544084397516569263302, 2351361747071896737244554098423912580225403854374905032, 9263404590590359981995351148904466350442217171390341542, 36499607266567782524974192112784638431298720410362605254, 143837062344333406631453749442723876962868730602068270746, 566914012007146007905297981645385065273277390673314295322, 2234735395419829179807297037353249598929082253109436158666
Generating function in Maple syntax:
(-(x^3+x^2-x+1)^2*(1-4*x)^(1/2)+3*x^6+6*x^5-x^4+2*x^3+5*x^2-4*x+1)/(2*x^7+8*x^6+6*x^5+8*x^3)
Generating function in latex syntax:
\frac{-\left(x^{3}+x^{2}-x +1\right)^{2} \sqrt{1-4 x}+3 x^{6}+6 x^{5}-x^{4}+2 x^{3}+5 x^{2}-4 x +1}{2 x^{7}+8 x^{6}+6 x^{5}+8 x^{3}}
Generating function in sympy syntax:
(3*x**6 + 6*x**5 - x**4 + 2*x**3 + 5*x**2 - 4*x - sqrt(1 - 4*x)*(x**3 + x**2 - x + 1)**2 + 1)/(2*x**7 + 8*x**6 + 6*x**5 + 8*x**3)
Implicit equation for the generating function in Maple syntax:
x^3*(x+2)*(x^3+2*x^2-x+2)*F(x)^2+(-3*x^6-6*x^5+x^4-2*x^3-5*x^2+4*x-1)*F(x)+x^6+2*x^5-x^4+x^3+3*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
x^{3} \left(x +2\right) \left(x^{3}+2 x^{2}-x +2\right) F \! \left(x \right)^{2}+\left(-3 x^{6}-6 x^{5}+x^{4}-2 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)+x^{6}+2 x^{5}-x^{4}+x^{3}+3 x^{2}-3 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) = 220
a(7) = 749
a(n+8) = 1/2*(1+2*n)/(11+n)*a(n)-3/2*(n+7)/(11+n)*a(3+n)+1/4*(20+19*n)/(11+n)*a(n+1)+1/4*(23+19*n)/(11+n)*a(n+2)+(22+5*n)/(11+n)*a(n+4)+1/4*(142+23*n)/(11+n)*a(n+5)-1/4*(157+23*n)/(11+n)*a(n+6)+(46+5*n)/(11+n)*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) = 20
a \! \left(5\right) = 66
a \! \left(6\right) = 220
a \! \left(7\right) = 749
a \! \left(n +8\right) = \frac{\left(1+2 n \right) a \! \left(n \right)}{22+2 n}-\frac{3 \left(n +7\right) a \! \left(3+n \right)}{2 \left(11+n \right)}+\frac{\left(20+19 n \right) a \! \left(n +1\right)}{44+4 n}+\frac{\left(23+19 n \right) a \! \left(n +2\right)}{44+4 n}+\frac{\left(22+5 n \right) a \! \left(n +4\right)}{11+n}+\frac{\left(142+23 n \right) a \! \left(n +5\right)}{44+4 n}-\frac{\left(157+23 n \right) a \! \left(n +6\right)}{4 \left(11+n \right)}+\frac{\left(46+5 n \right) a \! \left(n +7\right)}{11+n}, \quad n \geq 8
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21220/
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[29,x]+F[4,x]
F[4,x] = F[5,x,1]
F[5,x,k[0]] = F[6,x,k[0]]*F[9,x]
F[6,x,k[0]] = F[1,x]+F[10,x,k[0]]+F[27,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[9,x]
F[11,x,k[0]] = F[1,x]+F[12,x,k[0]]+F[16,x,k[0]]+F[21,x,k[0]]+F[4,x]
F[12,x,k[0]] = F[13,x,k[0]]*F[9,x]
F[13,x,k[0]] = 1/(-1+k[0])*(F[14,x,k[0]]*k[0]-F[14,x,1])
F[14,x,k[0]] = F[1,x]+F[15,x,k[0]]+F[16,x,k[0]]+F[4,x]
F[15,x,k[0]] = F[14,x,k[0]]*F[9,x]
F[16,x,k[0]] = F[17,x]*F[20,x,k[0]]
F[17,x] = F[18,x,1]
F[18,x,k[0]] = F[1,x]+F[19,x,k[0]]+F[5,x,k[0]]
F[19,x,k[0]] = F[18,x,k[0]]*F[20,x,k[0]]
F[20,x,k[0]] = k[0]*x
F[21,x,k[0]] = F[22,x,k[0]]
F[22,x,k[0]] = F[17,x]*F[23,x]*F[26,x,k[0]]*F[9,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[23,x]*F[9,x]
F[26,x,k[0]] = F[1,x]+F[20,x,k[0]]
F[27,x,k[0]] = F[20,x,k[0]]*F[28,x,k[0]]
F[28,x,k[0]] = 1/(-1+k[0])*(F[18,x,k[0]]*k[0]-F[18,x,1])
F[29,x] = F[30,x]*F[9,x]
F[30,x] = F[1,x]+F[31,x]+F[32,x]+F[4,x]
F[31,x] = F[15,x,1]
F[32,x] = F[33,x]
F[33,x] = F[17,x]*F[23,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_{9}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x , 1\right)
F_{5}\! \left(x , y\right) = F_{6}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{6}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{27}\! \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_{9}\! \left(x \right)
F_{11}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{21}\! \left(x , y\right)+F_{4}\! \left(x \right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{13}\! \left(x , y\right) = \frac{y F_{14}\! \left(x , y\right)-F_{14}\! \left(x , 1\right)}{-1+y}
F_{14}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{4}\! \left(x \right)
F_{15}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{16}\! \left(x , y\right) = F_{17}\! \left(x \right) F_{20}\! \left(x , y\right)
F_{17}\! \left(x \right) = F_{18}\! \left(x , 1\right)
F_{18}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{5}\! \left(x , y\right)
F_{19}\! \left(x , y\right) = F_{18}\! \left(x , y\right) F_{20}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = y x
F_{21}\! \left(x , y\right) = F_{22}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{17}\! \left(x \right) F_{23}\! \left(x \right) F_{26}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{23}\! \left(x \right) F_{9}\! \left(x \right)
F_{26}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{20}\! \left(x , y\right)
F_{27}\! \left(x , y\right) = F_{20}\! \left(x , y\right) F_{28}\! \left(x , y\right)
F_{28}\! \left(x , y\right) = \frac{y F_{18}\! \left(x , y\right)-F_{18}\! \left(x , 1\right)}{-1+y}
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{9}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x , 1\right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{17}\! \left(x \right) F_{23}\! \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)*F_9(x))
Eq(F_3(x), F_1(x) + F_29(x) + F_4(x))
Eq(F_4(x), F_5(x, 1))
Eq(F_5(x, k_0), F_6(x, k_0)*F_9(x))
Eq(F_6(x, k_0), F_1(x) + F_10(x, k_0) + F_27(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)*F_9(x))
Eq(F_11(x, k_0), F_1(x) + F_12(x, k_0) + F_16(x, k_0) + F_21(x, k_0) + F_4(x))
Eq(F_12(x, k_0), F_13(x, k_0)*F_9(x))
Eq(F_13(x, k_0), (k_0*F_14(x, k_0) - F_14(x, 1))/(k_0 - 1))
Eq(F_14(x, k_0), F_1(x) + F_15(x, k_0) + F_16(x, k_0) + F_4(x))
Eq(F_15(x, k_0), F_14(x, k_0)*F_9(x))
Eq(F_16(x, k_0), F_17(x)*F_20(x, k_0))
Eq(F_17(x), F_18(x, 1))
Eq(F_18(x, k_0), F_1(x) + F_19(x, k_0) + F_5(x, k_0))
Eq(F_19(x, k_0), F_18(x, k_0)*F_20(x, k_0))
Eq(F_20(x, k_0), k_0*x)
Eq(F_21(x, k_0), F_22(x, k_0))
Eq(F_22(x, k_0), F_17(x)*F_23(x)*F_26(x, k_0)*F_9(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_23(x)*F_9(x))
Eq(F_26(x, k_0), F_1(x) + F_20(x, k_0))
Eq(F_27(x, k_0), F_20(x, k_0)*F_28(x, k_0))
Eq(F_28(x, k_0), (k_0*F_18(x, k_0) - F_18(x, 1))/(k_0 - 1))
Eq(F_29(x), F_30(x)*F_9(x))
Eq(F_30(x), F_1(x) + F_31(x) + F_32(x) + F_4(x))
Eq(F_31(x), F_15(x, 1))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_17(x)*F_23(x)*F_9(x))
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, 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, 2, 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], [2, 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]]}, 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