0132_0213_0231_2013_3012
Counting sequence:
1, 1, 2, 6, 19, 59, 184, 585, 1904, 6337, 21507, 74206, 259591, 918666, 3282879, 11829031, 42927279, 156746679, 575454968, 2122750310, 7863827578, 29243272288, 109122241692, 408467597244, 1533343878176, 5771055545984, 21772768933242, 82325530739886, 311922512552878, 1184095643906610, 4502958414510650, 17152566388214123, 65438629899106384, 250018089900859713, 956539358027609659, 3664323520515818202, 14054411674742799439, 53967263306209626630, 207453277168966964075, 798284681371496109264, 3074823386972472305017, 11854586955596485256414, 45744308071301441692609, 176666408889903797599410, 682842168541192716137223, 2641316288466276940791080, 10224440274193952534295365, 39606276189064577200342216, 153525657763000577371176859, 595495321420476517769412318, 2311238965752634318846406187, 8975730613517833297023881450, 34877327844620418180722301509, 135598716719727504537905751988, 527470304093003974178991671359, 2052872975186486606857589373378, 7993551801984580479174802702509, 31140358759700525122177331222960, 121368608840489898623940826936007, 473239830879019571501706295360114, 1846043427155406337886379306446965, 7204139509149202921403977616773824, 28125228813188890357845769708308627, 109844474010503028774357023426360411, 429163990253017883438395439282424575, 1677359704398497344153932755996435527, 6558162836142078467160960646195891768, 25649957565516425493143836326402779262, 100354114019319820941696853303747565562, 392756687241263392793728462348172739224, 1537615879722237534947568529330940389460, 6021492319728707773862984283549732003544, 23587871164903233242097905246627449990624, 92426850346759218127951110825698372544996, 362267161508669731850571455276520182183742, 1420292972913344792424385123298905985186846, 5569830008827016004152592796474988069263346, 21848316301399233977806152145325546178418498, 85724143552159049345501620376648696043813214, 336429993363768142022258393311151195374372548, 1320656766101754256552813709620943256640545098, 5185448736381651946299097161502692869282239600, 20364857897955558166126267784504365596045764676, 79996813997343217994592968020745215112183914270, 314309830846421087526518468971929262867012596460, 1235193469659415257584488025596560682904863505622, 4855138412090084766530550372678178272503772395028, 19087795925478110279307564455111708916811003762282, 75057741709414986763106962182949549367728580158392, 295201675079799416973815763657874048693884262616974, 1161245090144455231569186554254564664674345448111760, 4568871152212289981869299480965818018275476995881214, 17979273265746716656669847896027722103488196564304404, 70763918168386148273615493257698833434308356008394466, 278564963659349899542503856626589955693144337699287272, 1096767099743065298894192966991833318766121960546868616, 4318909515188474442916909272943862483957728377226133136, 17009990902638026786900500980487887139374092880991039408, 67004335366164286249582522572917812507147952462164043834, 263978877223250039740287191483986484913907537155695391558, 1040163354925482345323440763090584661054585381404986852238
Generating function in Maple syntax:
1/4*((-x^4-3*x^3+6*x^2-4*x+1)*(1-4*x)^(1/2)-2*x^5-x^4+5*x^3-6*x^2+4*x-1)/x/(x-1/2)/(x-1)^2
Generating function in latex syntax:
\frac{\left(-x^{4}-3 x^{3}+6 x^{2}-4 x +1\right) \sqrt{1-4 x}-2 x^{5}-x^{4}+5 x^{3}-6 x^{2}+4 x -1}{4 x \left(x -\frac{1}{2}\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**5/2 - x**4/4 + 5*x**3/4 - 3*x**2/2 + x + sqrt(1 - 4*x)*(-x**4 - 3*x**3 + 6*x**2 - 4*x + 1)/4 - 1/4)/(x*(x - 1)**2*(x - 1/2))
Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x-1)^4*F(x)^2+(x^4+x^3-2*x^2+2*x-1)*(2*x-1)^2*(x-1)^2*F(x)+(x^3+2*x^2-3*x+1)*(x^6+4*x^4-10*x^3+10*x^2-5*x+1) = 0
Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(x^{4}+x^{3}-2 x^{2}+2 x -1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+\left(x^{3}+2 x^{2}-3 x +1\right) \left(x^{6}+4 x^{4}-10 x^{3}+10 x^{2}-5 x +1\right) = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 59
a(6) = 184
a(7) = 585
a(8) = 1904
a(9) = 6337
a(n+7) = -4*(2*n-1)/(n+8)*a(n)-2*(5*n+24)/(n+8)*a(1+n)+(221+83*n)/(n+8)*a(n+2)-(136*n+467)/(n+8)*a(n+3)+(492+109*n)/(n+8)*a(n+4)-16*(3*n+17)/(n+8)*a(n+5)+(75+11*n)/(n+8)*a(n+6)+2/(n+8), n >= 10
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) = 59
a \! \left(6\right) = 184
a \! \left(7\right) = 585
a \! \left(8\right) = 1904
a \! \left(9\right) = 6337
a \! \left(n +7\right) = -\frac{4 \left(2 n -1\right) a \! \left(n \right)}{n +8}-\frac{2 \left(5 n +24\right) a \! \left(1+n \right)}{n +8}+\frac{\left(221+83 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(136 n +467\right) a \! \left(n +3\right)}{n +8}+\frac{\left(492+109 n \right) a \! \left(n +4\right)}{n +8}-\frac{16 \left(3 n +17\right) a \! \left(n +5\right)}{n +8}+\frac{\left(75+11 n \right) a \! \left(n +6\right)}{n +8}+\frac{2}{n +8}, \quad n \geq 10
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21588/
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]
F[5,x] = F[23,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[10,x]*F[9,x]
F[9,x] = x
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[9,x]
F[14,x] = F[15,x]+F[18,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[9,x]
F[17,x] = F[1,x]+F[15,x]
F[18,x] = F[19,x]+F[20,x]+F[22,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[9,x]
F[21,x] = F[12,x]+F[18,x]
F[22,x] = F[14,x]*F[9,x]
F[23,x] = F[24,x,1]
F[24,x,k[0]] = F[1,x]+F[25,x,k[0]]+F[27,x,k[0]]
F[25,x,k[0]] = F[24,x,k[0]]*F[26,x,k[0]]
F[26,x,k[0]] = k[0]*x
F[27,x,k[0]] = F[28,x,k[0]]*F[9,x]
F[28,x,k[0]] = 1/(-1+k[0])*(F[24,x,k[0]]*k[0]-F[24,x,1])
F[29,x] = F[30,x]*F[9,x]
F[30,x] = F[1,x]+F[31,x]+F[33,x]+F[35,x]
F[31,x] = F[32,x]*F[9,x]
F[32,x] = F[28,x,1]
F[33,x] = F[34,x]
F[34,x] = F[17,x]*F[23,x]*F[9,x]
F[35,x] = F[36,x]*F[9,x]
F[36,x] = F[37,x,1]
F[37,x,k[0]] = F[1,x]+F[33,x]+F[38,x,k[0]]+F[41,x,k[0]]+F[42,x,k[0]]
F[38,x,k[0]] = F[39,x,k[0]]*F[9,x]
F[39,x,k[0]] = (F[40,x,k[0]]*k[0]-F[40,x,1])/(-1+k[0])
F[40,x,k[0]] = 1/(-1+k[0])*(F[24,x,k[0]]*k[0]-F[24,x,1])
F[41,x,k[0]] = F[26,x,k[0]]*F[28,x,k[0]]
F[42,x,k[0]] = F[43,x,k[0]]*F[9,x]
F[43,x,k[0]] = (F[37,x,k[0]]*k[0]-F[37,x,1])/(-1+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_{1}\! \left(x \right)+F_{29}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{23}\! \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_{10}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{9}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{9}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x , 1\right)
F_{24}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{27}\! \left(x , y\right)
F_{25}\! \left(x , y\right) = F_{24}\! \left(x , y\right) F_{26}\! \left(x , y\right)
F_{26}\! \left(x , y\right) = y x
F_{27}\! \left(x , y\right) = F_{28}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{28}\! \left(x , y\right) = \frac{y F_{24}\! \left(x , y\right)-F_{24}\! \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_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{9}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x , 1\right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{17}\! \left(x \right) F_{23}\! \left(x \right) F_{9}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{9}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x , 1\right)
F_{37}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{33}\! \left(x \right)+F_{38}\! \left(x , y\right)+F_{41}\! \left(x , y\right)+F_{42}\! \left(x , y\right)
F_{38}\! \left(x , y\right) = F_{39}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{39}\! \left(x , y\right) = \frac{y F_{40}\! \left(x , y\right)-F_{40}\! \left(x , 1\right)}{-1+y}
F_{40}\! \left(x , y\right) = \frac{y F_{24}\! \left(x , y\right)-F_{24}\! \left(x , 1\right)}{-1+y}
F_{41}\! \left(x , y\right) = F_{26}\! \left(x , y\right) F_{28}\! \left(x , y\right)
F_{42}\! \left(x , y\right) = F_{43}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{43}\! \left(x , y\right) = \frac{y F_{37}\! \left(x , y\right)-F_{37}\! \left(x , 1\right)}{-1+y}
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))
Eq(F_5(x), F_23(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_10(x)*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_9(x))
Eq(F_14(x), F_15(x) + F_18(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_9(x))
Eq(F_17(x), F_1(x) + F_15(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_22(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_9(x))
Eq(F_21(x), F_12(x) + F_18(x))
Eq(F_22(x), F_14(x)*F_9(x))
Eq(F_23(x), F_24(x, 1))
Eq(F_24(x, k_0), F_1(x) + F_25(x, k_0) + F_27(x, k_0))
Eq(F_25(x, k_0), F_24(x, k_0)*F_26(x, k_0))
Eq(F_26(x, k_0), k_0*x)
Eq(F_27(x, k_0), F_28(x, k_0)*F_9(x))
Eq(F_28(x, k_0), (-k_0*F_24(x, k_0) + F_24(x, 1))/(1 - k_0))
Eq(F_29(x), F_30(x)*F_9(x))
Eq(F_30(x), F_1(x) + F_31(x) + F_33(x) + F_35(x))
Eq(F_31(x), F_32(x)*F_9(x))
Eq(F_32(x), F_28(x, 1))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_17(x)*F_23(x)*F_9(x))
Eq(F_35(x), F_36(x)*F_9(x))
Eq(F_36(x), F_37(x, 1))
Eq(F_37(x, k_0), F_1(x) + F_33(x) + F_38(x, k_0) + F_41(x, k_0) + F_42(x, k_0))
Eq(F_38(x, k_0), F_39(x, k_0)*F_9(x))
Eq(F_39(x, k_0), (-k_0*F_40(x, k_0) + F_40(x, 1))/(1 - k_0))
Eq(F_40(x, k_0), (k_0*F_24(x, k_0) - F_24(x, 1))/(k_0 - 1))
Eq(F_41(x, k_0), F_26(x, k_0)*F_28(x, k_0))
Eq(F_42(x, k_0), F_43(x, k_0)*F_9(x))
Eq(F_43(x, k_0), (-k_0*F_37(x, k_0) + F_37(x, 1))/(1 - k_0))
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, 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]]}, {"patt": [3, 0, 1, 2], "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], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[2, 0], [2, 0], 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