0231_1302_2031_2103_3012
Counting sequence:
1, 1, 2, 6, 19, 54, 144, 369, 914, 2206, 5223, 12186, 28112, 64285, 145994, 329754, 741571, 1661858, 3713680, 8279701, 18425026, 40938546, 90846215, 201385534, 446042008, 987220209, 2183716490, 4827987990, 10669889803, 23572593350, 52063478664, 114962942081, 253803973802, 560233321598, 1236463071967, 2728630996170, 6020991574520, 13284876846845, 29310212454850, 64663171135770, 142651232160907, 314686272235874, 694171078760952, 1531242361338133, 3377628925180266, 7450271194198530, 16433333916735855, 36247146120997982, 79949804230618384, 176342581701743361, 388950039005025522, 857882491839405062, 1892167543783600867, 4173395444054551286, 9204876285432614880, 20302293316018309489, 44778668500427080610, 98763475817476272766, 217831567107865741111, 480446073828578856154, 1059663479275138661376, 2337172974727962919805, 5154818599267249521562, 11369349558662707062778, 25075961997959011085619, 55306907957926719917378, 121983469624016298520896, 269043460664138842015861, 593394858216593237685746, 1308775078555238993141106, 2886597098621189115172151, 6366595457733968807889758, 14041977769642782454903400, 30970574296647929497695793, 68307783887665606461840954, 150657618815970555355844822, 332285946694962554317921691, 732879925151596567842040006, 1616417925030539617870076568, 3565122635307797083741118785, 7863126738104327808581829242, 17342674238039463528804482718, 38250476329075884802061427823, 84364088993082663440444328138, 186070869875520785391914873832, 410392248545949177256563897501, 905148645798955300634206223570, 1996368271304553385993855878586, 4403128993165983952929111035067, 9711407003686947492191426332482, 21419182962076522659094070520680, 47241496174284055849218447473077, 104194401664174184343145876691546, 229807990542707117364721936690818, 506857485080864668328617184451679, 1117909386211271249922188777392190, 2465626789424065949900198999656864, 5438111112594347007491878087341697, 11994131700909499765881715895631202, 26453890355876766754693364146233222, 58345892127156466730246973582254387
Generating function in Maple syntax:
(x^8+x^7+x^6-4*x^5+4*x^4-x^3+5*x^2-4*x+1)/(x^3+x^2+x-1)/(x^3+2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{8}+x^{7}+x^{6}-4 x^{5}+4 x^{4}-x^{3}+5 x^{2}-4 x +1}{\left(x^{3}+x^{2}+x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**8 + x**7 + x**6 - 4*x**5 + 4*x**4 - x**3 + 5*x**2 - 4*x + 1)/((x - 1)**2*(x**3 + 2*x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3+x^2+x-1)*(x^3+2*x-1)*(x-1)^2*F(x)-x^8-x^7-x^6+4*x^5-4*x^4+x^3-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+x^{2}+x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)-x^{8}-x^{7}-x^{6}+4 x^{5}-4 x^{4}+x^{3}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
3/2-12897/2596*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^6+_Z^5+3*_Z^4+_Z^2-3*_Z+1))-4975/649*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^6+_Z^5+3*_Z^4+_Z^2-3*_Z+1))-48247/2596*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^6+_Z^5+3*_Z^4+_Z^2-3*_Z+1))-24813/2596*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^6+_Z^5+3*_Z^4+_Z^2-3*_Z+1))-6103/649*Sum(_alpha^(-n),_alpha = RootOf(_Z^6+_Z^5+3*_Z^4+_Z^2-3*_Z+1))+24807/2596*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^6+_Z^5+3*_Z^4+_Z^2-3*_Z+1))+n+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
\frac{3}{2}-\frac{12897 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{2596}-\frac{4975 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{649}-\frac{48247 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{2596}-\frac{24813 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{2596}-\frac{6103 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{649}+\frac{24807 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{2596}+n +\left(\left\{\begin{array}{cc}1 & 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) = 19
a(5) = 54
a(6) = 144
a(7) = 369
a(8) = 914
a(n+2) = -1/3*a(n)-1/3*a(n+1)-1/3*a(n+4)+a(n+5)-1/3*a(n+6)+4/3*n+8/3, n >= 9
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) = 144
a \! \left(7\right) = 369
a \! \left(8\right) = 914
a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{3}-\frac{a \! \left(n +1\right)}{3}-\frac{a \! \left(n +4\right)}{3}+a \! \left(n +5\right)-\frac{a \! \left(n +6\right)}{3}+\frac{4 n}{3}+\frac{8}{3}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21083/
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[29,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[22,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[4,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[15,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[7,x]
F[29,x] = F[30,x]+F[42,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[14,x]+F[33,x]
F[33,x] = F[30,x]+F[34,x]
F[34,x] = F[35,x]+F[36,x]+F[40,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[21,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[30,x]+F[39,x]
F[42,x] = F[35,x]+F[43,x]+F[66,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[25,x]+F[45,x]
F[45,x] = F[46,x]+F[65,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[30,x]+F[46,x]
F[50,x] = F[51,x]+F[58,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[41,x]+F[54,x]
F[54,x] = F[51,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[39,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[46,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[46,x]
F[65,x] = F[63,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[29,x]+F[68,x]
F[68,x] = F[51,x]+F[69,x]
F[69,x] = F[59,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_{29}\! \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_{22}\! \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_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \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_{19}\! \left(x \right)
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_{21}\! \left(x \right) = F_{15}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{42}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{21}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{30}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{35}\! \left(x \right)+F_{43}\! \left(x \right)+F_{66}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{25}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{65}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \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_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{30}\! \left(x \right)+F_{46}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{58}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{41}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{51}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \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_{39}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{46}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{46}\! \left(x \right)
F_{65}\! \left(x \right) = F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{29}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{51}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{59}\! \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_29(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_22(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_4(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_15(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_7(x))
Eq(F_29(x), F_30(x) + F_42(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_14(x) + F_33(x))
Eq(F_33(x), F_30(x) + F_34(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_40(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_21(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_30(x) + F_39(x))
Eq(F_42(x), F_35(x) + F_43(x) + F_66(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_25(x) + F_45(x))
Eq(F_45(x), F_46(x) + F_65(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_30(x) + F_46(x))
Eq(F_50(x), F_51(x) + F_58(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_41(x) + F_54(x))
Eq(F_54(x), F_51(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_39(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_46(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_46(x))
Eq(F_65(x), F_63(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_29(x) + F_68(x))
Eq(F_68(x), F_51(x) + F_69(x))
Eq(F_69(x), F_59(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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