0231_0321_1302_2013_2103
Counting sequence:
1, 1, 2, 6, 19, 55, 153, 424, 1182, 3306, 9250, 25869, 72327, 202212, 565365, 1580740, 4419705, 12357338, 34550616, 96602095, 270095524, 755176196, 2111442250, 5903507455, 16505968782, 46150023003, 129033603173, 360772751131, 1008706064021, 2820301478723, 7885449204670, 22047397991853, 61643635714083, 172353119650586, 481892372330408, 1347351640520867, 3767140854366295, 10532773917243859, 29449211134007092, 82338806778670426, 230215983405623938, 643674606044299260, 1799688242046538977, 5031855751565336041, 14068865769645011386, 39335981358908219835, 109981817639263564310, 307504727055625020195, 859770816588111866474, 2403884532555080250118, 6721164215353827796645, 18792104112312541764712, 52541965298403758117108, 146905216197152085203440, 410741060471669061133265, 1148415441770146551287136, 3210923264846294408844179, 8977611966658596688972829, 25101041032741654926677543, 70181498517348476806054473, 196224639755612805269305549, 548636179914312023477624840, 1533964635052206826632574429, 4288903261827095541764939409, 11991600568213266691276214918, 33528031622311253407765885414, 93743024383791426035396789451, 262101715949596518150351875595, 732825828431465024031252421497, 2048951464779945580857979260056, 5728785670682057250056641612438, 16017453719498807149798397993494, 44784154689058355789582661508130, 125214690570447561985554776732561, 350095225499110447498109572786923, 978852132756062259208225777221128, 2736831090555750204645114141394345, 7652069365311589396473363638667720, 21394877372446796889254860269702145, 59819214375796098496414004074889022, 167252111159364811473130720156369084, 467630158288788854317043361451615735, 1307475065189653731542237308646252276, 3655647557780007432688165436981510304, 10221043156004259099438065166842274910, 28577624496258501240050290868056530715, 79901885686628994669656562724717689798, 223402450302158158798526130925687321867, 624624242245638266456924190989852610837, 1746424193079536403391835379814882377699, 4882931618549116428967601927497004122369, 13652479899160916354576205358177909644807, 38171783255972705592930312850931969966430, 106726766690241447900847163525052499998417, 298403735863478523799905818886531424773627, 834324812216225988991566511136572507379254, 2332738530452278476308160858434159916908232, 6522242862467306489228387867591996713307123, 18235928031230318010677567364526476997332435, 50986919403735804033578919578283683225601571, 142557370583550155507975240456763624937576228
Generating function in Maple syntax:
-(x-1)*(x^2+x-1)/(x^5+3*x^4-x^2+3*x-1)
Generating function in latex syntax:
-\frac{\left(x -1\right) \left(x^{2}+x -1\right)}{x^{5}+3 x^{4}-x^{2}+3 x -1}
Generating function in sympy syntax:
(1 - x)*(x**2 + x - 1)/(x**5 + 3*x**4 - x**2 + 3*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^5+3*x^4-x^2+3*x-1)*F(x)+(x-1)*(x^2+x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}+3 x^{4}-x^{2}+3 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{2}+x -1\right) = 0
Explicit closed form in Maple syntax:
-1186/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 1)^(-n+3)-1186/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 2)^(-n+3)-1186/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 3)^(-n+3)-1186/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 4)^(-n+3)-1186/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 5)^(-n+3)-2697/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 3)^(-n+2)-2697/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 4)^(-n+2)-2697/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 5)^(-n+2)-2697/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 1)^(-n+2)-2697/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 2)^(-n+2)+1694/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 1)^(-n+1)+1694/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 2)^(-n+1)+1694/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 3)^(-n+1)+1694/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 4)^(-n+1)+1694/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 5)^(-n+1)+2205/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 1)^(-n-1)+2205/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 2)^(-n-1)+2205/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 3)^(-n-1)+2205/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 4)^(-n-1)+2205/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 5)^(-n-1)+7799/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 1)^(-n)+7799/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 2)^(-n)+7799/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 3)^(-n)+7799/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 4)^(-n)+7799/44719*RootOf(_Z^5+3*_Z^4-_Z^2+3*_Z-1,index = 5)^(-n)
Explicit closed form in latex syntax:
-\frac{1186 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{44719}-\frac{1186 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{44719}-\frac{1186 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{44719}-\frac{1186 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{44719}-\frac{1186 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{44719}-\frac{2697 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{44719}-\frac{2697 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{44719}-\frac{2697 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{44719}-\frac{2697 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{44719}-\frac{2697 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{44719}+\frac{1694 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{44719}+\frac{1694 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{44719}+\frac{1694 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{44719}+\frac{1694 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{44719}+\frac{1694 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{44719}+\frac{2205 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{44719}+\frac{2205 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{44719}+\frac{2205 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{44719}+\frac{2205 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{44719}+\frac{2205 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{44719}+\frac{7799 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{44719}+\frac{7799 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{44719}+\frac{7799 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{44719}+\frac{7799 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{44719}+\frac{7799 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{44719}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+1) = -1/3*a(n)+1/3*a(n+3)-a(n+4)+1/3*a(n+5), 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) = 19
a \! \left(n +1\right) = -\frac{a \! \left(n \right)}{3}+\frac{a \! \left(n +3\right)}{3}-a \! \left(n +4\right)+\frac{a \! \left(n +5\right)}{3}, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20888/
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[17,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[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[41,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[26,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[11,x]
F[26,x] = F[27,x]+F[38,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[2,x]+F[27,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[2,x]+F[32,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[27,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[32,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[17,x]+F[43,x]
F[43,x] = F[32,x]+F[44,x]
F[44,x] = F[36,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_{17}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{41}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{38}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{2}\! \left(x \right)+F_{27}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{27}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{17}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{32}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{36}\! \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_17(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_6(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_41(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_26(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_11(x))
Eq(F_26(x), F_27(x) + F_38(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_2(x) + F_27(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_2(x) + F_32(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_27(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_32(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_17(x) + F_43(x))
Eq(F_43(x), F_32(x) + F_44(x))
Eq(F_44(x), F_36(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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