0123_0132_0213_1302_2103
Counting sequence:
1, 1, 2, 6, 19, 54, 150, 418, 1171, 3282, 9193, 25742, 72082, 201849, 565236, 1582818, 4432317, 12411676, 34756027, 97326218, 272539561, 763183974, 2137120095, 5984510227, 16758235880, 46927561156, 131409774329, 367982660103, 1030450275271, 2885537512811, 8080279987862, 22626954039573, 63361548099393, 177429351318168, 496849834852355, 1391312973635859, 3896049982954057, 10909986291587823, 30550891647540541, 85550701487078975, 239564940014458301, 670846170592755478, 1878549442885103713, 5260442951095294420, 14730653029408000485, 41249784607516069218, 115510475114004616884, 323460352290710610074, 905775856265572331719, 2536415656458160912367, 7102645028374235385628, 19889313595207021888571, 55695419622995647974473, 155962132736903086198072, 436735857499498097070732, 1222977692588902748585944, 3424665987202105480352812, 9589984506644135470287302, 26854532144552500055821420, 75199902169097891700291103, 210579922070585314995008117, 589680336013470903990774739, 1651263307830482574023453169, 4623980732036785863533889341, 12948387885115186885272092114, 36258963550989105825599216636, 101534835800160386157100687371, 284324808856382875826834871064, 796185824245859085110255340561, 2229534134850178667617865866907, 6243294350499719571169179851016, 17482900906381963404421528184661, 48956817818129397908125627351478, 137092237936473943818526393445092, 383895084281213204990299016908332, 1075009336440848631304355819599064, 3010314850993125501306718993005348, 8429690045402122973884052340333087, 23605396039589853968076718611933610, 66101448473756243729871425664393176, 185101808205229774054365831034754898, 518334774682716975574502075092352776, 1451476575244995799090808246033191001, 4064524225245250988232400332732911340, 11381759416142850633048942876243011171, 31871983097637905892902159460207489027, 89250112345141044758955823213362015757, 249924283946129558487365704250720843382, 699855115749735609581311352438308737315, 1959782280087076951614989777688142222293, 5487916711487942118607868182568039577749, 15367640649802416640565002006858715451372, 43033521016653376638970513836725465004480, 120505416107225134253123420412012767019310, 337447528533764040376609704048203160033064, 944943706200091292549983939616805206011033, 2646096155354777042473642409367590037596527, 7409779881533704746494681272790519675825728, 20749373669461472054326695383305307058037124, 58103818812202951843357818731293384584844910, 162706297276342391574098441994860867024946793
Generating function in Maple syntax:
(x^2+x-1)^2/(x^7+3*x^6+x^5-2*x^4+2*x^3-3*x+1)
Generating function in latex syntax:
\frac{\left(x^{2}+x -1\right)^{2}}{x^{7}+3 x^{6}+x^{5}-2 x^{4}+2 x^{3}-3 x +1}
Generating function in sympy syntax:
(x**2 + x - 1)**2/(x**7 + 3*x**6 + x**5 - 2*x**4 + 2*x**3 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^7+3*x^6+x^5-2*x^4+2*x^3-3*x+1)*F(x)-(x^2+x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}+3 x^{6}+x^{5}-2 x^{4}+2 x^{3}-3 x +1\right) F \! \left(x \right)-\left(x^{2}+x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
-466486/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 1)^(-n+5)-466486/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 2)^(-n+5)-466486/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 3)^(-n+5)-466486/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 4)^(-n+5)-466486/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 5)^(-n+5)-466486/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 6)^(-n+5)-466486/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 7)^(-n+5)-1727025/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 1)^(-n+4)-1727025/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 2)^(-n+4)-1727025/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 3)^(-n+4)-1727025/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 4)^(-n+4)-1727025/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 5)^(-n+4)-1727025/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 6)^(-n+4)-1727025/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 7)^(-n+4)-1916037/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 1)^(-n+3)-1916037/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 2)^(-n+3)-1916037/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 3)^(-n+3)-1916037/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 4)^(-n+3)-1916037/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 5)^(-n+3)-1916037/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 6)^(-n+3)-1916037/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 7)^(-n+3)-825374/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 1)^(-n+2)-825374/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 2)^(-n+2)-825374/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 3)^(-n+2)-825374/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 4)^(-n+2)-825374/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 5)^(-n+2)-825374/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 6)^(-n+2)-825374/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 7)^(-n+2)-549613/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 1)^(-n+1)-549613/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 2)^(-n+1)-549613/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 3)^(-n+1)-549613/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 4)^(-n+1)-549613/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 5)^(-n+1)-549613/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 6)^(-n+1)-549613/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 7)^(-n+1)+815483/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 1)^(-n-1)+815483/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 2)^(-n-1)+815483/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 3)^(-n-1)+815483/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 4)^(-n-1)+815483/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 5)^(-n-1)+815483/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 6)^(-n-1)+815483/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 7)^(-n-1)+1203468/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 1)^(-n)+1203468/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 2)^(-n)+1203468/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 3)^(-n)+1203468/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 4)^(-n)+1203468/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 5)^(-n)+1203468/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 6)^(-n)+1203468/9893333*RootOf(_Z^7+3*_Z^6+_Z^5-2*_Z^4+2*_Z^3-3*_Z+1,index = 7)^(-n)
Explicit closed form in latex syntax:
-\frac{466486 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{9893333}-\frac{466486 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{9893333}-\frac{466486 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{9893333}-\frac{466486 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{9893333}-\frac{466486 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{9893333}-\frac{466486 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{9893333}-\frac{466486 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{9893333}-\frac{1727025 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{9893333}-\frac{1727025 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{9893333}-\frac{1727025 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{9893333}-\frac{1727025 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{9893333}-\frac{1727025 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{9893333}-\frac{1727025 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{9893333}-\frac{1727025 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{9893333}-\frac{1916037 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{9893333}-\frac{1916037 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{9893333}-\frac{1916037 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{9893333}-\frac{1916037 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{9893333}-\frac{1916037 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{9893333}-\frac{1916037 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{9893333}-\frac{1916037 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{9893333}-\frac{825374 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{9893333}-\frac{825374 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{9893333}-\frac{825374 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{9893333}-\frac{825374 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{9893333}-\frac{825374 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{9893333}-\frac{825374 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{9893333}-\frac{825374 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{9893333}-\frac{549613 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{9893333}-\frac{549613 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{9893333}-\frac{549613 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{9893333}-\frac{549613 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{9893333}-\frac{549613 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{9893333}-\frac{549613 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{9893333}-\frac{549613 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{9893333}+\frac{815483 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{9893333}+\frac{815483 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{9893333}+\frac{815483 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{9893333}+\frac{815483 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{9893333}+\frac{815483 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{9893333}+\frac{815483 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{9893333}+\frac{815483 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{9893333}+\frac{1203468 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{9893333}+\frac{1203468 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{9893333}+\frac{1203468 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{9893333}+\frac{1203468 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{9893333}+\frac{1203468 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{9893333}+\frac{1203468 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{9893333}+\frac{1203468 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-2 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{9893333}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 150
a(n+4) = -1/2*a(n)-3/2*a(n+1)-1/2*a(n+2)+a(n+3)+3/2*a(n+6)-1/2*a(n+7), 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) = 19
a \! \left(5\right) = 54
a \! \left(6\right) = 150
a \! \left(n +4\right) = -\frac{a \! \left(n \right)}{2}-\frac{3 a \! \left(n +1\right)}{2}-\frac{a \! \left(n +2\right)}{2}+a \! \left(n +3\right)+\frac{3 a \! \left(n +6\right)}{2}-\frac{a \! \left(n +7\right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16010/
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[14,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[4,x]*F[7,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[52,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[31,x]
F[19,x] = F[20,x]+F[7,x]
F[20,x] = F[16,x]+F[21,x]+F[25,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[4,x]
F[24,x] = F[12,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[25,x]
F[29,x] = F[30,x]
F[30,x] = F[28,x]*F[4,x]
F[31,x] = F[32,x]+F[41,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[2,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[2,x]*F[4,x]
F[38,x] = F[32,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[32,x]*F[4,x]
F[41,x] = 2*F[16,x]+F[42,x]+F[46,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[36,x]
F[45,x] = F[39,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[38,x]+F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[46,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[49,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[62,x]
F[54,x] = F[2,x]+F[55,x]
F[55,x] = F[16,x]+F[37,x]+F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[60,x]
F[58,x] = F[4,x]+F[59,x]
F[59,x] = F[13,x]+F[16,x]+F[21,x]
F[60,x] = F[36,x]+F[61,x]
F[61,x] = 2*F[16,x]+F[40,x]+F[42,x]
F[62,x] = F[32,x]+F[63,x]
F[63,x] = F[40,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_{14}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{52}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{31}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{41}\! \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_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{36}\! \left(x \right)
F_{45}\! \left(x \right) = F_{39}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{38}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{46}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \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_{62}\! \left(x \right)
F_{54}\! \left(x \right) = F_{2}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{16}\! \left(x \right)+F_{37}\! \left(x \right)+F_{56}\! \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_{4}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{13}\! \left(x \right)+F_{16}\! \left(x \right)+F_{21}\! \left(x \right)
F_{60}\! \left(x \right) = F_{36}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{62}\! \left(x \right) = F_{32}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{40}\! \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_14(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_4(x)*F_7(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_52(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_31(x))
Eq(F_19(x), F_20(x) + F_7(x))
Eq(F_20(x), F_16(x) + F_21(x) + F_25(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_4(x))
Eq(F_24(x), F_12(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_25(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_28(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_41(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_2(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_2(x)*F_4(x))
Eq(F_38(x), F_32(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_32(x)*F_4(x))
Eq(F_41(x), 2*F_16(x) + F_42(x) + F_46(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_36(x))
Eq(F_45(x), F_39(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_38(x) + F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_46(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_49(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_62(x))
Eq(F_54(x), F_2(x) + F_55(x))
Eq(F_55(x), F_16(x) + F_37(x) + F_56(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_4(x) + F_59(x))
Eq(F_59(x), F_13(x) + F_16(x) + F_21(x))
Eq(F_60(x), F_36(x) + F_61(x))
Eq(F_61(x), 2*F_16(x) + F_40(x) + F_42(x))
Eq(F_62(x), F_32(x) + F_63(x))
Eq(F_63(x), F_40(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 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