0123_0321_1032_1302_2103
Counting sequence:
1, 1, 2, 6, 19, 48, 116, 294, 773, 2008, 5143, 13173, 33914, 87417, 224998, 578649, 1488633, 3831019, 9858993, 25368501, 65274861, 167963082, 432205008, 1112146365, 2861740993, 7363750786, 18948256343, 48757299453, 125461246513, 322834073418, 830709618431, 2137564379808, 5500335174518, 14153344268641, 36419080969324, 93712798052883, 241139764426550, 620495672004474, 1596646152188950, 4108455631406418, 10571789927581722, 27203103140716354, 69998441612681379, 180118488753548187, 463477032476828340, 1192609160483014452, 3068796315585282321, 7896560866813179546, 20319261075260870254, 52285086839119437940, 134538864167463753695, 346192520002513694706, 890815168144736048545, 2292226486556408699739, 5898308036918849204492, 15177399747546726888663, 39054159541159215793196, 100493325789398930252083, 258587270776427728630831, 665391219589430952459213, 1712170416499319244116519, 4405720197125856977160123, 11336704727703983301617515, 29171365482308647985784072, 75063132060137600045984094, 193150841639371940394336475, 497011603460785410022212339, 1278899599287629645262408564, 3290837022051759265494424941, 8467911250999514935608341115, 21789447631197928128878407371, 56068139355697195955681205816, 144273333772300494223229323645, 371241048427239990721458968686, 955269504306809317501463196992, 2458079002105358560577109921111, 6325076173111753651338794251006, 16275550363271545642063995067653, 41879897154988509592934012523788, 107764453217535208969696421622036, 277297179940448707640457000797468, 713535156604066554068480691514804, 1836053362747248756608684888404708, 4724493137660580719196236832659220, 12156964421994638283060955666796789, 31282040136653735836806318246221781, 80494274815904932771754228168273974, 207126141704117316052842813221688546, 532972545877973348165990362978878207, 1371433525109697826847881113560694052, 3528943335526774618295252576600265672, 9080601310488392466313534203529695954, 23366008552737186527423270030894131977, 60124912108625854425988335247018486476, 154712134420001653145393943514445035947, 398101946387023338758070912716128919553, 1024387390887465925134257226623712022414, 2635931665576592866899402539935149570805, 6782722832589686234935772224434869505554, 17453156932909370778694412519113196874381, 44910089125440408837523440387122060249797
Generating function in Maple syntax:
-1/(x+1)/(x^6+5*x^5+4*x^4+4*x^3-x^2+2*x-1)
Generating function in latex syntax:
-\frac{1}{\left(x +1\right) \left(x^{6}+5 x^{5}+4 x^{4}+4 x^{3}-x^{2}+2 x -1\right)}
Generating function in sympy syntax:
-1/((x + 1)*(x**6 + 5*x**5 + 4*x**4 + 4*x**3 - x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x+1)*(x^6+5*x^5+4*x^4+4*x^3-x^2+2*x-1)*F(x)+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x +1\right) \left(x^{6}+5 x^{5}+4 x^{4}+4 x^{3}-x^{2}+2 x -1\right) F \! \left(x \right)+1 = 0
Explicit closed form in Maple syntax:
2577157/228980896*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 1)^(-n+5)+2577157/228980896*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 2)^(-n+5)+2577157/228980896*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 3)^(-n+5)+2577157/228980896*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 4)^(-n+5)+2577157/228980896*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 5)^(-n+5)+2577157/228980896*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 6)^(-n+5)+40294879/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 1)^(-n+4)+40294879/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 2)^(-n+4)+40294879/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 3)^(-n+4)+40294879/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 4)^(-n+4)+40294879/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 5)^(-n+4)+40294879/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 6)^(-n+4)+3862385/57245224*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 1)^(-n+3)+3862385/57245224*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 2)^(-n+3)+3862385/57245224*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 3)^(-n+3)+3862385/57245224*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 4)^(-n+3)+3862385/57245224*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 5)^(-n+3)+3862385/57245224*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 6)^(-n+3)+20702951/171735672*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 1)^(-n+2)+20702951/171735672*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 2)^(-n+2)+20702951/171735672*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 3)^(-n+2)+20702951/171735672*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 4)^(-n+2)+20702951/171735672*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 5)^(-n+2)+20702951/171735672*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 6)^(-n+2)+86665025/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 1)^(-n+1)+86665025/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 2)^(-n+1)+86665025/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 3)^(-n+1)+86665025/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 4)^(-n+1)+86665025/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 5)^(-n+1)+86665025/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 6)^(-n+1)+34553597/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 1)^(-n-1)+34553597/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 2)^(-n-1)+34553597/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 3)^(-n-1)+34553597/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 4)^(-n-1)+34553597/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 5)^(-n-1)+34553597/686942688*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 6)^(-n-1)+69029933/343471344*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 1)^(-n)+69029933/343471344*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 2)^(-n)+69029933/343471344*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 3)^(-n)+1/8*(-1)^(-n)+69029933/343471344*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 4)^(-n)+69029933/343471344*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 5)^(-n)+69029933/343471344*RootOf(_Z^6+5*_Z^5+4*_Z^4+4*_Z^3-_Z^2+2*_Z-1,index = 6)^(-n)
Explicit closed form in latex syntax:
\frac{2577157 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{228980896}+\frac{2577157 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{228980896}+\frac{2577157 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{228980896}+\frac{2577157 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{228980896}+\frac{2577157 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{228980896}+\frac{2577157 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{228980896}+\frac{40294879 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{686942688}+\frac{40294879 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{686942688}+\frac{40294879 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{686942688}+\frac{40294879 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{686942688}+\frac{40294879 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{686942688}+\frac{40294879 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{686942688}+\frac{3862385 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{57245224}+\frac{3862385 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{57245224}+\frac{3862385 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{57245224}+\frac{3862385 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{57245224}+\frac{3862385 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{57245224}+\frac{3862385 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{57245224}+\frac{20702951 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{171735672}+\frac{20702951 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{171735672}+\frac{20702951 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{171735672}+\frac{20702951 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{171735672}+\frac{20702951 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{171735672}+\frac{20702951 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{171735672}+\frac{86665025 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{686942688}+\frac{86665025 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{686942688}+\frac{86665025 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{686942688}+\frac{86665025 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{686942688}+\frac{86665025 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{686942688}+\frac{86665025 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{686942688}+\frac{34553597 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{686942688}+\frac{34553597 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{686942688}+\frac{34553597 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{686942688}+\frac{34553597 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{686942688}+\frac{34553597 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{686942688}+\frac{34553597 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{686942688}+\frac{69029933 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{343471344}+\frac{69029933 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{343471344}+\frac{69029933 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{343471344}+\frac{\left(-1\right)^{-n}}{8}+\frac{69029933 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{343471344}+\frac{69029933 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{343471344}+\frac{69029933 \mathit{RootOf}\! \left(\textit{\_Z}^{6}+5 \textit{\_Z}^{5}+4 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-\textit{\_Z}^{2}+2 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{343471344}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 48
a(6) = 116
a(n+7) = a(n)+6*a(n+1)+9*a(n+2)+8*a(n+3)+3*a(n+4)+a(n+5)+a(n+6), 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) = 48
a \! \left(6\right) = 116
a \! \left(n +7\right) = a \! \left(n \right)+6 a \! \left(n +1\right)+9 a \! \left(n +2\right)+8 a \! \left(n +3\right)+3 a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17720/
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[25,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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]+F[19,x]+F[23,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[4,x]
F[21,x] = F[22,x]
F[22,x] = x^2
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[15,x]
F[25,x] = F[2,x]+F[26,x]
F[26,x] = F[18,x]+F[27,x]+F[60,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[40,x]
F[29,x] = F[15,x]+F[30,x]
F[30,x] = F[18,x]+F[31,x]+F[37,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[35,x]
F[33,x] = F[34,x]+F[4,x]
F[34,x] = F[22,x]
F[35,x] = F[36,x]
F[36,x] = F[15,x]*F[4,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[15,x]+F[39,x]
F[39,x] = F[18,x]+F[19,x]+F[36,x]
F[40,x] = F[41,x]+F[46,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[2,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[2,x]*F[4,x]
F[46,x] = 2*F[18,x]+F[47,x]+F[54,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[44,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[44,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[41,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[41,x]+F[56,x]
F[56,x] = 2*F[18,x]+F[53,x]+F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[44,x]+F[59,x]
F[59,x] = F[51,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[71,x]
F[62,x] = F[2,x]+F[63,x]
F[63,x] = F[18,x]+F[64,x]+F[70,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[68,x]
F[66,x] = F[4,x]+F[67,x]
F[67,x] = F[18,x]+F[31,x]+F[36,x]
F[68,x] = F[44,x]+F[69,x]
F[69,x] = 2*F[18,x]+F[47,x]+F[53,x]
F[70,x] = F[4,x]*F[43,x]
F[71,x] = F[41,x]+F[72,x]
F[72,x] = 2*F[18,x]+F[57,x]+F[73,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[41,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_{25}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
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_{22}\! \left(x \right) = x^{2}
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)
F_{25}\! \left(x \right) = F_{2}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \left(x \right)+F_{60}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{40}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{37}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{22}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{36}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{2}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{54}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{44}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{41}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{53}\! \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_{44}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{51}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{71}\! \left(x \right)
F_{62}\! \left(x \right) = F_{2}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{70}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{68}\! \left(x \right) = F_{44}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{53}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \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_{18}\! \left(x \right)+F_{57}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{41}\! \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_25(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_14(x))
Eq(F_10(x), F_1(x) + F_11(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_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_23(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), x**2)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_15(x))
Eq(F_25(x), F_2(x) + F_26(x))
Eq(F_26(x), F_18(x) + F_27(x) + F_60(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_40(x))
Eq(F_29(x), F_15(x) + F_30(x))
Eq(F_30(x), F_18(x) + F_31(x) + F_37(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_35(x))
Eq(F_33(x), F_34(x) + F_4(x))
Eq(F_34(x), F_22(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_15(x)*F_4(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_15(x) + F_39(x))
Eq(F_39(x), F_18(x) + F_19(x) + F_36(x))
Eq(F_40(x), F_41(x) + F_46(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_2(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_2(x)*F_4(x))
Eq(F_46(x), 2*F_18(x) + F_47(x) + F_54(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_44(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_44(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_41(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_41(x) + F_56(x))
Eq(F_56(x), 2*F_18(x) + F_53(x) + F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_44(x) + F_59(x))
Eq(F_59(x), F_51(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_71(x))
Eq(F_62(x), F_2(x) + F_63(x))
Eq(F_63(x), F_18(x) + F_64(x) + F_70(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_68(x))
Eq(F_66(x), F_4(x) + F_67(x))
Eq(F_67(x), F_18(x) + F_31(x) + F_36(x))
Eq(F_68(x), F_44(x) + F_69(x))
Eq(F_69(x), 2*F_18(x) + F_47(x) + F_53(x))
Eq(F_70(x), F_4(x)*F_43(x))
Eq(F_71(x), F_41(x) + F_72(x))
Eq(F_72(x), 2*F_18(x) + F_57(x) + F_73(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_41(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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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": 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