0213_0231_2130
Counting sequence:
1, 1, 2, 6, 21, 76, 271, 938, 3146, 10252, 32583, 101368, 309697, 931708, 2766374, 8121630, 23612985, 68076148, 194824915, 553979438, 1566301826, 4406286736, 12340335627, 34422720496, 95676632581, 265071811456, 732238423646, 2017390937058, 5544712079421, 15205817438572, 41616185589463, 113686083626738, 310032947795642, 844150921347988, 2295061672117839, 6231259081990120, 16896825988348105, 45763484053346308, 123808637685586070, 334603868389703718, 903413648518040001, 2436930498380215396, 6567853201671534427, 17686773570315372086, 47592449734524417266, 127970427323349651736, 343858506235619773971, 923344468401889628128, 2477853778237558814221, 6645514989112052144392, 17812908437453488346318, 47720567534317772807658, 127776655868702413565637, 341965642888501624107868, 914761169943033828675679, 2445884377487076691798010, 6536950723112184932811050, 17463577819790146095152668, 46635554581903825407084375, 124489792497766890360332248, 332192172943015775460008401, 886115074249804901457353356, 2362879752465495056142000902, 6298675961363957880778676526, 16784876800512603447813264201, 44714988743555147233153848916, 119085463823596136309611046371, 317058491913164824333709838398, 843915905707439025053772104546, 2245649818670668094362295895712, 5974089439480116270594267434523, 15888825579073693228502785202128, 42247952657102774522056474433557, 112309521412561848226104274376848, 298488513325884502680971089496894, 793125234869214011056321427992050, 2106986938628272778030862273434445, 5596165607114649933606322876753228, 14860400214402987778283074384532455, 39453376006549467865744554763051778, 104725860387945883005997158272304026, 277934261941054615210684030867646116, 737480963216208287265369647445048927, 1956510684291859997201935495372111816, 5189653221682025515974468302608364761, 13763253320340646929674296700209630996, 36494917626284487839246588415402857846, 96755127880180810092933032063187032118, 256476540093168434164965730647333325521, 679759086316109484440467496925931710532, 1801348426530086209907023680421796804203, 4772844722389183509467296568479833699398, 12644313620321877530756140050308834216402, 33492921248543436461311637725571088357928, 88705797575583163958295454403406817997859, 234905688719180868654146366298647524450240, 621983572854846834108284678670308476507165, 1646680725423527317477801858978258769918488, 4358993385878321322633062170020158620027310, 11537468084022988476560342319438748798424058, 30533982039166674789282609738489669310764501
Generating function in Maple syntax:
-(7*x^5-34*x^4+49*x^3-31*x^2+9*x-1)/(2*x-1)^2/(x^2-3*x+1)^2
Generating function in latex syntax:
-\frac{7 x^{5}-34 x^{4}+49 x^{3}-31 x^{2}+9 x -1}{\left(2 x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(-7*x**5 + 34*x**4 - 49*x**3 + 31*x**2 - 9*x + 1)/((2*x - 1)**2*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x^2-3*x+1)^2*F(x)+7*x^5-34*x^4+49*x^3-31*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+7 x^{5}-34 x^{4}+49 x^{3}-31 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
-1/8*(((6/5*n-222/25)*5^(1/2)+14/5*n-94/5)*(3/2-1/2*5^(1/2))^(-n)+(4/5*n+72/25*5^(1/2)+4/5)*(3/2+1/2*5^(1/2))^(-n)+2^n*(3+5^(1/2))*(n+8))*(5^(1/2)-3)
Explicit closed form in latex syntax:
-\frac{\left(\left(\left(\frac{6 n}{5}-\frac{222}{25}\right) \sqrt{5}+\frac{14 n}{5}-\frac{94}{5}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\left(\frac{4 n}{5}+\frac{72 \sqrt{5}}{25}+\frac{4}{5}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}+2^{n} \left(3+\sqrt{5}\right) \left(n +8\right)\right) \left(\sqrt{5}-3\right)}{8}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 76
a(n+6) = -4*a(n)+28*a(n+1)-69*a(n+2)+74*a(n+3)-39*a(n+4)+10*a(n+5), n >= 6
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) = 76
a \! \left(n +6\right) = -4 a \! \left(n \right)+28 a \! \left(n +1\right)-69 a \! \left(n +2\right)+74 a \! \left(n +3\right)-39 a \! \left(n +4\right)+10 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/313/
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[3,x] = F[14,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[14,x]*F[7,x]
F[7,x] = F[42,x]+F[8,x]
F[8,x] = F[37,x]*F[9,x]
F[9,x] = F[10,x]+F[36,x]
F[10,x] = F[11,x]*F[15,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[14,x]
F[14,x] = x
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[14,x]*F[17,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[19,x]+F[25,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[21,x]
F[21,x] = F[11,x]+F[22,x]
F[22,x] = F[19,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[14,x]*F[22,x]
F[25,x] = F[26,x]+F[27,x]+F[35,x]
F[26,x] = 0
F[27,x] = F[14,x]*F[28,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[12,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[14,x]*F[29,x]
F[32,x] = F[25,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[14,x]*F[32,x]
F[35,x] = F[14,x]*F[18,x]
F[36,x] = F[35,x]
F[37,x] = F[1,x]+F[38,x]
F[38,x] = F[14,x]*F[39,x]
F[39,x] = F[37,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[38,x]
F[42,x] = F[11,x]*F[43,x]
F[43,x] = -F[58,x]+F[44,x]
F[44,x] = F[45,x]+F[47,x]
F[45,x] = 1/F[14,x]*F[46,x]
F[46,x] = F[2,x]
F[47,x] = F[48,x]*F[49,x]
F[48,x] = F[16,x]
F[49,x] = F[37,x]+F[50,x]
F[50,x] = F[41,x]+F[51,x]
F[51,x] = F[26,x]+F[52,x]+F[55,x]
F[52,x] = F[14,x]*F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[41,x]+F[51,x]
F[55,x] = F[14,x]*F[56,x]
F[56,x] = F[40,x]+F[57,x]
F[57,x] = F[51,x]
F[58,x] = F[15,x]*F[37,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_{3}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{42}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{37}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \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_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{25}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right) F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{35}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{14}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right) F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{25}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right) F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right) F_{18}\! \left(x \right)
F_{36}\! \left(x \right) = F_{35}\! \left(x \right)
F_{37}\! \left(x \right) = F_{1}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{38}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = -F_{58}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = \frac{F_{46}\! \left(x \right)}{F_{14}\! \left(x \right)}
F_{46}\! \left(x \right) = F_{2}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right) F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{16}\! \left(x \right)
F_{49}\! \left(x \right) = F_{37}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{41}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{26}\! \left(x \right)+F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{14}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{41}\! \left(x \right)+F_{51}\! \left(x \right)
F_{55}\! \left(x \right) = F_{14}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{40}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{51}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right) F_{37}\! \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))
Eq(F_3(x), F_14(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_14(x)*F_7(x))
Eq(F_7(x), F_42(x) + F_8(x))
Eq(F_8(x), F_37(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_36(x))
Eq(F_10(x), F_11(x)*F_15(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_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_14(x)*F_17(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_19(x) + F_25(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_21(x))
Eq(F_21(x), F_11(x) + F_22(x))
Eq(F_22(x), F_19(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_14(x)*F_22(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_35(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_14(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_12(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_14(x)*F_29(x))
Eq(F_32(x), F_25(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_14(x)*F_32(x))
Eq(F_35(x), F_14(x)*F_18(x))
Eq(F_36(x), F_35(x))
Eq(F_37(x), F_1(x) + F_38(x))
Eq(F_38(x), F_14(x)*F_39(x))
Eq(F_39(x), F_37(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_38(x))
Eq(F_42(x), F_11(x)*F_43(x))
Eq(F_43(x), F_44(x) - F_58(x))
Eq(F_44(x), F_45(x) + F_47(x))
Eq(F_45(x), F_46(x)/F_14(x))
Eq(F_46(x), F_2(x))
Eq(F_47(x), F_48(x)*F_49(x))
Eq(F_48(x), F_16(x))
Eq(F_49(x), F_37(x) + F_50(x))
Eq(F_50(x), F_41(x) + F_51(x))
Eq(F_51(x), F_26(x) + F_52(x) + F_55(x))
Eq(F_52(x), F_14(x)*F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_41(x) + F_51(x))
Eq(F_55(x), F_14(x)*F_56(x))
Eq(F_56(x), F_40(x) + F_57(x))
Eq(F_57(x), F_51(x))
Eq(F_58(x), F_15(x)*F_37(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "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.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "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:
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