0132_0213_1302_3102
Counting sequence:
1, 1, 2, 6, 20, 65, 203, 614, 1814, 5268, 15104, 42883, 120821, 338314, 942536, 2614782, 7227812, 19917077, 54734495, 150054830, 410491418, 1120769736, 3054676232, 8312221831, 22585495145, 61284769810, 166084729868, 449572466034, 1215617502764, 3283614722393, 8861212315859, 23891577390518, 64362065482526, 173248432336764, 465995588869136, 1252520519273227, 3364298019133277, 9030783208704922, 24226539978599120, 64953874991497830, 174051677057754548, 466145825383464221, 1247803077196919687, 3338570296440534206, 8928410654964305570, 23866862207226383280, 63772272541173968408, 170330040202852382479, 454757997056501614289, 1213684295555264861218, 3237975739201639647764, 8635545055869337538778, 23022884839535588304956, 61360483280830413217889, 163486460483155440116411, 435455209665242380750022, 1159520205270391520536742, 3086672200419185535080292, 8214555733041906074220320, 21855606197583103792427155, 58134037083253580609120453, 154593216451881998810337706, 411003972103844134766187992, 1092447067666070825393997006, 2903063973905717129885396420, 7712896714125785005158354725, 20487355003378559410594867823, 54408202936044267342318298574, 144462628880727071401576201514, 383496774275574315028657225368, 1017853590541441661250500955944, 2701024596491863686960076960663, 7166276099620468714185641986361, 19010010804990239475848001035314, 50419321721935725649197199518236, 133702443476772076638291549375042, 354495910647299262588047643168332, 939754505161204480443544366744937, 2490867352976186995407554931922595, 6601177581472006720524857286121430, 17491555726376132061876300861171758, 46341830574685078073760245921066316, 122760068593677753362211926191476272, 325148432017013215557738142604360731, 861089265292753879832975527597798445, 2280121420555550424142630278991907770, 6036877128618993555038038541460939488, 15981314305337581024820954686138256694, 42301876675247434301973193273715204564, 111957944043910282631780424997748834093, 296278029539107725436637445417403757399, 783960738497702897204802866218157144798, 2074151893644091922409603981895916946866, 5487053471580246250183015265413722017664, 14514136400842957042364984853381335239736, 38388180841040644199390276845619514656287, 101521753572806247428936744536559017211297, 268458297118862942624177174251314134972482, 709825442057699938660421248783872180214820, 1876653724634484296430180200228113799215530, 4961070514312538247551316633548617833578588
Generating function in Maple syntax:
(x^6-5*x^5+18*x^4-32*x^3+24*x^2-8*x+1)/(2*x-1)/(x-1)/(x^2-3*x+1)^2
Generating function in latex syntax:
\frac{x^{6}-5 x^{5}+18 x^{4}-32 x^{3}+24 x^{2}-8 x +1}{\left(2 x -1\right) \left(x -1\right) \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(x**6 - 5*x**5 + 18*x**4 - 32*x**3 + 24*x**2 - 8*x + 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)*(x^2-3*x+1)^2*F(x)-x^6+5*x^5-18*x^4+32*x^3-24*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)-x^{6}+5 x^{5}-18 x^{4}+32 x^{3}-24 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/50*((-5*n+6)*5^(1/2)+15*n)*(3/2-1/2*5^(1/2))^(-n)+1/50*((5*n-6)*5^(1/2)+15*n)*(3/2+1/2*5^(1/2))^(-n)-2^(-1+n)+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-5 n +6\right) \sqrt{5}+15 n \right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\\\frac{\left(\left(5 n -6\right) \sqrt{5}+15 n \right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}-2^{-1+n}+1 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(6) = 203
a(n+5) = 2*a(n)-13*a(n+1)+28*a(n+2)-23*a(n+3)+8*a(n+4)-1, 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) = 20
a \! \left(5\right) = 65
a \! \left(6\right) = 203
a \! \left(n +5\right) = 2 a \! \left(n \right)-13 a \! \left(n +1\right)+28 a \! \left(n +2\right)-23 a \! \left(n +3\right)+8 a \! \left(n +4\right)-1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15288/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[16,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[4,x]
F[16,x] = F[17,x]+F[30,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[6,x]
F[20,x] = F[17,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[17,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[24,x]*F[4,x]
F[27,x] = F[21,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[27,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]+F[83,x]
F[31,x] = 0
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[50,x]
F[34,x] = F[35,x]+F[7,x]
F[35,x] = F[31,x]+F[36,x]+F[44,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[11,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[38,x]*F[4,x]
F[41,x] = F[14,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[41,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[13,x]+F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[44,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[46,x]
F[50,x] = F[30,x]+F[51,x]
F[51,x] = 2*F[31,x]+F[52,x]+F[72,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[69,x]
F[54,x] = F[55,x]+F[67,x]
F[55,x] = F[31,x]+F[56,x]+F[65,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[60,x]
F[58,x] = F[11,x]+F[59,x]
F[59,x] = F[15,x]+F[31,x]+F[36,x]
F[60,x] = F[55,x]+F[61,x]
F[61,x] = 2*F[31,x]+F[52,x]+F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[21,x]+F[64,x]
F[64,x] = F[62,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[17,x]+F[55,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[54,x]
F[69,x] = F[64,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[69,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[63,x]+F[74,x]
F[74,x] = F[75,x]+F[81,x]
F[75,x] = F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[27,x]+F[78,x]
F[78,x] = F[75,x]+F[79,x]
F[79,x] = F[80,x]
F[80,x] = F[4,x]*F[78,x]
F[81,x] = F[82,x]
F[82,x] = F[4,x]*F[74,x]
F[83,x] = F[4,x]*F[84,x]
F[84,x] = F[63,x]+F[66,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{16}\! \left(x \right)+F_{6}\! \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_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{30}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{6}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{17}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{21}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{83}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{50}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)+F_{44}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{11}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{13}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{44}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{50}\! \left(x \right) = F_{30}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{52}\! \left(x \right)+F_{72}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{69}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{67}\! \left(x \right)
F_{55}\! \left(x \right) = F_{31}\! \left(x \right)+F_{56}\! \left(x \right)+F_{65}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{11}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{60}\! \left(x \right) = F_{55}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{52}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{21}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{17}\! \left(x \right)+F_{55}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{69}\! \left(x \right) = F_{64}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{63}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{81}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{27}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{75}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{63}\! \left(x \right)+F_{66}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_16(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_30(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x) + F_6(x))
Eq(F_20(x), F_17(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_17(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_24(x)*F_4(x))
Eq(F_27(x), F_21(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_27(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_83(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_50(x))
Eq(F_34(x), F_35(x) + F_7(x))
Eq(F_35(x), F_31(x) + F_36(x) + F_44(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_11(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_38(x)*F_4(x))
Eq(F_41(x), F_14(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_41(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_13(x) + F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_44(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_46(x))
Eq(F_50(x), F_30(x) + F_51(x))
Eq(F_51(x), 2*F_31(x) + F_52(x) + F_72(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_69(x))
Eq(F_54(x), F_55(x) + F_67(x))
Eq(F_55(x), F_31(x) + F_56(x) + F_65(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_60(x))
Eq(F_58(x), F_11(x) + F_59(x))
Eq(F_59(x), F_15(x) + F_31(x) + F_36(x))
Eq(F_60(x), F_55(x) + F_61(x))
Eq(F_61(x), 2*F_31(x) + F_52(x) + F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_21(x) + F_64(x))
Eq(F_64(x), F_62(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_17(x) + F_55(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_54(x))
Eq(F_69(x), F_64(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_69(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_63(x) + F_74(x))
Eq(F_74(x), F_75(x) + F_81(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_27(x) + F_78(x))
Eq(F_78(x), F_75(x) + F_79(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_4(x)*F_78(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_4(x)*F_74(x))
Eq(F_83(x), F_4(x)*F_84(x))
Eq(F_84(x), F_63(x) + F_66(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": 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:
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