0123_0231_2031_2103_3012
Counting sequence:
1, 1, 2, 6, 19, 50, 117, 271, 630, 1466, 3410, 7929, 18434, 42855, 99627, 231606, 538420, 1251676, 2909795, 6764454, 15725449, 36557235, 84985262, 197566766, 459287010, 1067712761, 2482131030, 5770254579, 13414214439, 31184265190, 72494621272, 168529547876, 391783666275, 910786524346, 2117321788365, 4922175982679, 11442670895654, 26600982509970, 61839781721282, 143760051039561, 334201572186090, 776924396200431, 1806130095268675, 4198743065591254, 9760893402436844, 22691324171396700, 52750928774907667, 122631031384366446, 285082560774680705, 662736548330216891, 1540675554825655710, 3581636128591214054, 8326293824452547634, 19356284771000870505, 44997902792688730302, 104607432660516997531, 243182777167174402495, 565331368973177942726, 1314235985245702020720, 3055227994957924579204, 7102543383355547638899, 16511410145396495779010, 38384371664436316638437, 89232838085871505996191, 207441181074138380490710, 482242238715108446118186, 1121077192082920083369330, 2606188279892681738362329, 6058652694227313494466514, 14084658714979497090044215, 32742859036376546019561947, 76117912373397957373063926, 176952677762420277170112100, 411365067576841462783770396, 956307759579081791384150915, 2223145821345982725755024054, 5168187012456626057280540729, 12014577154256994511715724995, 27930503259203714146341117582, 64930542481553779472872443486, 150945198080510904637650820290, 350905012537628869113548691481, 815755183933418577538216877350, 1896400724805508899025204069379, 4408596819087318411112727144919, 10248744191584356012825990173350, 23825439661383940115277720299592, 55387427420070426731293907696996, 128760147129027755976152272665155, 299331026208326354581146722901066, 695860211786993978522429531069885, 1617678730073356982381147420072679, 3760646792854409344679729920979334, 8742443130203508047799324453862530, 20323714534975062436419660939701602, 47246904137372580558340063832359081, 109835726472371124849980194071536570, 255337085677343275869680115489593151, 593586708224660158467420022158065395, 1379921679791665048512880029566546454, 3207928708603018104473480159873101724
Generating function in Maple syntax:
(2*x^7-4*x^6-x^5+3*x^4+3*x^2-3*x+1)/(x-1)/(x^3-2*x^2+3*x-1)
Generating function in latex syntax:
\frac{2 x^{7}-4 x^{6}-x^{5}+3 x^{4}+3 x^{2}-3 x +1}{\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(2*x**7 - 4*x**6 - x**5 + 3*x**4 + 3*x**2 - 3*x + 1)/((x - 1)*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(x^3-2*x^2+3*x-1)*F(x)+2*x^7-4*x^6-x^5+3*x^4+3*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+2 x^{7}-4 x^{6}-x^{5}+3 x^{4}+3 x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
-134/23*Sum(_alpha^(2-n),_alpha = RootOf(_Z^4-3*_Z^3+5*_Z^2-4*_Z+1))+233/23*Sum(_alpha^(1-n),_alpha = RootOf(_Z^4-3*_Z^3+5*_Z^2-4*_Z+1))-170/23*Sum(_alpha^(-n),_alpha = RootOf(_Z^4-3*_Z^3+5*_Z^2-4*_Z+1))+48/23*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^4-3*_Z^3+5*_Z^2-4*_Z+1))-piecewise(n = 0,14,n = 1,5,Or(n = 2,n = 3),-2,0)
Explicit closed form in latex syntax:
-\frac{134 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{2-n}\right)}{23}+\frac{233 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{23}-\frac{170 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{23}+\frac{48 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{23}-\left(\left\{\begin{array}{cc}14 & n =0 \\ 5 & n =1 \\ -2 & n =2\text{ or } n =3 \\ 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) = 50
a(6) = 117
a(7) = 271
a(n+3) = a(n)-2*a(n+1)+3*a(n+2)+1, n >= 8
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) = 50
a \! \left(6\right) = 117
a \! \left(7\right) = 271
a \! \left(n +3\right) = a \! \left(n \right)-2 a \! \left(n +1\right)+3 a \! \left(n +2\right)+1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17580/
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[18,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[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]
F[17,x] = F[11,x]
F[18,x] = F[19,x]+F[32,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[10,x]+F[22,x]
F[22,x] = F[19,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[31,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[19,x]+F[28,x]
F[31,x] = F[22,x]*F[4,x]
F[32,x] = F[24,x]+F[33,x]+F[49,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[36,x]+F[7,x]
F[36,x] = F[37,x]
F[37,x] = F[17,x]*F[4,x]
F[38,x] = F[39,x]+F[47,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[30,x]+F[42,x]
F[42,x] = F[39,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[28,x]
F[47,x] = F[48,x]
F[48,x] = F[27,x]*F[4,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[59,x]
F[51,x] = F[19,x]+F[52,x]
F[52,x] = F[24,x]+F[31,x]+F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[57,x]
F[55,x] = F[11,x]+F[56,x]
F[56,x] = F[37,x]
F[57,x] = F[28,x]+F[58,x]
F[58,x] = F[48,x]
F[59,x] = F[60,x]+F[67,x]
F[60,x] = F[24,x]+F[25,x]+F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[22,x]+F[63,x]
F[63,x] = F[60,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[27,x]
F[67,x] = 2*F[24,x]+F[65,x]+F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[56,x]
F[71,x] = F[72,x]
F[72,x] = F[4,x]*F[46,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_{18}\! \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_{14}\! \left(x \right)+F_{7}\! \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_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{32}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{31}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{19}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{24}\! \left(x \right)+F_{33}\! \left(x \right)+F_{49}\! \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_{36}\! \left(x \right)+F_{7}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{47}\! \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_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{39}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{28}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{59}\! \left(x \right)
F_{51}\! \left(x \right) = F_{19}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{11}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{37}\! \left(x \right)
F_{57}\! \left(x \right) = F_{28}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{48}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{67}\! \left(x \right)
F_{60}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{22}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{60}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{27}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{56}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \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_18(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_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_19(x) + F_32(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_10(x) + F_22(x))
Eq(F_22(x), F_19(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_31(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_19(x) + F_28(x))
Eq(F_31(x), F_22(x)*F_4(x))
Eq(F_32(x), F_24(x) + F_33(x) + F_49(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_36(x) + F_7(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_17(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_47(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_42(x))
Eq(F_42(x), F_39(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_28(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_27(x)*F_4(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_59(x))
Eq(F_51(x), F_19(x) + F_52(x))
Eq(F_52(x), F_24(x) + F_31(x) + F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_57(x))
Eq(F_55(x), F_11(x) + F_56(x))
Eq(F_56(x), F_37(x))
Eq(F_57(x), F_28(x) + F_58(x))
Eq(F_58(x), F_48(x))
Eq(F_59(x), F_60(x) + F_67(x))
Eq(F_60(x), F_24(x) + F_25(x) + F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_22(x) + F_63(x))
Eq(F_63(x), F_60(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_27(x))
Eq(F_67(x), 2*F_24(x) + F_65(x) + F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_56(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_4(x)*F_46(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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