0132_0321_1302_2103
Counting sequence:
1, 1, 2, 6, 20, 62, 180, 510, 1448, 4146, 11928, 34338, 98756, 283798, 815372, 2342806, 6732352, 19347258, 55599632, 159778394, 459155900, 1319476126, 3791783812, 10896480238, 31313320216, 89985380994, 258591777544, 743117380242, 2135502691828, 6136812272870, 17635410004988, 50679029035078, 145636760202672, 418517605196810, 1202697625465728, 3456202464311242, 9932118618730604, 28542014333649774, 82021431027543156, 235705688774820766, 677349456391602184, 1946504933565212882, 5593687897219945336, 16074628813987588866, 46193798484262147876, 132747514303271521846, 381477668689446687468, 1096255643448173893878, 3150319230796768683040, 9053099352548755513818, 26015969139227531091696, 74762313313473426193722, 214845100025664613401628, 617402203855012500062270, 1774234000586886753256804, 5098631441843355933950798, 14651980838578582164199928, 42105522813874663194629154, 120999001497582506394389992, 347715866826561090830185330, 999234064302277701024679252, 2871507487922831918811979910, 8251875659337545089520674268, 23713485750449615305857294758, 68145647080911102590858866064, 195830729608701217169640786090, 562760444750677610104148440160, 1617209509502377788699990331050, 4647388817072279201907665036492, 13355241043378691967836466592270, 38379070559262991900104963732052, 110290263740552489029765623996158, 316942075425659187354596315649768, 910799156409933078670362702687346, 2617371335765179124628127361090136, 7521562422486325623239208074120162, 21614778347383095327866965813676164, 62114573643605291994494377245735894, 178499182222425174381565811850495180, 512954628601153450661703940215698006, 1474082109101622121169142858008304128, 4236082380812345535335482380263223418, 12173266215112659293347330017481671632, 34982419372964478837916186506682688538, 100529278137916326125136018082430800508, 288891847507290563653889417248818403550, 830190976221660874097773942027395666756, 2385726918038007275054256744626124529326, 6855883875484870929026050963958565676760, 19701812206062655909214130761906631813570, 56617266460848184912128957951486397407432, 162701523492964845188843558086647142747218, 467556761420784723059932243799100692037812, 1343621869402748360990801556255243488503846, 3861177672741666534289951693961162704089020, 11095899344884730239694247541527234066756358, 31886380971531818192659021980684696009439600, 91632166069566020892674844162818350806184266, 263324140362522979818111347473752271138594112, 756716837240537614311025842646503994388755850, 2174583655623011804840882639472745296933211436
Generating function in Maple syntax:
(x-1)^4/(2*x^4-9*x^3+9*x^2-5*x+1)
Generating function in latex syntax:
\frac{\left(x -1\right)^{4}}{2 x^{4}-9 x^{3}+9 x^{2}-5 x +1}
Generating function in sympy syntax:
(x - 1)**4/(2*x**4 - 9*x**3 + 9*x**2 - 5*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^4-9*x^3+9*x^2-5*x+1)*F(x)-(x-1)^4 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{4}-9 x^{3}+9 x^{2}-5 x +1\right) F \! \left(x \right)-\left(x -1\right)^{4} = 0
Explicit closed form in Maple syntax:
-48/1027*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4-9*_Z^3+9*_Z^2-5*_Z+1))+81/1027*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4-9*_Z^3+9*_Z^2-5*_Z+1))+76/1027*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-9*_Z^3+9*_Z^2-5*_Z+1))+77/1027*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-9*_Z^3+9*_Z^2-5*_Z+1))+piecewise(n = 0,1/2,0)
Explicit closed form in latex syntax:
-\frac{48 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+9 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{1027}+\frac{81 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+9 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{1027}+\frac{76 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+9 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{1027}+\frac{77 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-9 \textit{\_Z}^{3}+9 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{1027}+\left(\left\{\begin{array}{cc}\frac{1}{2} & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+4) = -2*a(n)+9*a(n+1)-9*a(n+2)+5*a(n+3), n >= 5
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(n +4\right) = -2 a \! \left(n \right)+9 a \! \left(n +1\right)-9 a \! \left(n +2\right)+5 a \! \left(n +3\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15453/
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[20,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[11,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[16,x]
F[19,x] = F[13,x]*F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[15,x]+F[22,x]+F[63,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[36,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[15,x]+F[26,x]+F[34,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[17,x]+F[28,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[11,x]+F[29,x]
F[32,x] = F[33,x]
F[33,x] = F[28,x]*F[4,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[13,x]
F[36,x] = F[37,x]+F[49,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[41,x]+F[44,x]
F[44,x] = 2*F[15,x]+F[45,x]+F[48,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[41,x]+F[47,x]
F[47,x] = F[45,x]
F[48,x] = F[4,x]*F[43,x]
F[49,x] = 2*F[15,x]+F[50,x]+F[61,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[41,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[52,x]
F[55,x] = F[56,x]+F[59,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[41,x]+F[56,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[55,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[43,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[74,x]
F[65,x] = F[2,x]+F[66,x]
F[66,x] = F[15,x]+F[67,x]+F[73,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]+F[71,x]
F[69,x] = F[11,x]+F[70,x]
F[70,x] = F[15,x]+F[19,x]+F[26,x]
F[71,x] = F[41,x]+F[72,x]
F[72,x] = 2*F[15,x]+F[48,x]+F[50,x]
F[73,x] = F[4,x]*F[65,x]
F[74,x] = F[41,x]+F[75,x]
F[75,x] = 2*F[15,x]+F[45,x]+F[76,x]
F[76,x] = F[4,x]*F[74,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_{20}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{63}\! \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_{36}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{34}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{17}\! \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_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{49}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{61}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{41}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{41}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{43}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{74}\! \left(x \right)
F_{65}\! \left(x \right) = F_{2}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{15}\! \left(x \right)+F_{67}\! \left(x \right)+F_{73}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{71}\! \left(x \right)
F_{69}\! \left(x \right) = F_{11}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{71}\! \left(x \right) = F_{41}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{74}\! \left(x \right) = F_{41}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \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_20(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_11(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_13(x)*F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_15(x) + F_22(x) + F_63(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_36(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_15(x) + F_26(x) + F_34(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_17(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_11(x) + F_29(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_28(x)*F_4(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_13(x))
Eq(F_36(x), F_37(x) + F_49(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), F_41(x) + F_44(x))
Eq(F_44(x), 2*F_15(x) + F_45(x) + F_48(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_41(x) + F_47(x))
Eq(F_47(x), F_45(x))
Eq(F_48(x), F_4(x)*F_43(x))
Eq(F_49(x), 2*F_15(x) + F_50(x) + F_61(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_41(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_52(x))
Eq(F_55(x), F_56(x) + F_59(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_41(x) + F_56(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_55(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_43(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_74(x))
Eq(F_65(x), F_2(x) + F_66(x))
Eq(F_66(x), F_15(x) + F_67(x) + F_73(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_71(x))
Eq(F_69(x), F_11(x) + F_70(x))
Eq(F_70(x), F_15(x) + F_19(x) + F_26(x))
Eq(F_71(x), F_41(x) + F_72(x))
Eq(F_72(x), 2*F_15(x) + F_48(x) + F_50(x))
Eq(F_73(x), F_4(x)*F_65(x))
Eq(F_74(x), F_41(x) + F_75(x))
Eq(F_75(x), 2*F_15(x) + F_45(x) + F_76(x))
Eq(F_76(x), F_4(x)*F_74(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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