0132_0321_1230_2130_2301
Counting sequence:
1, 1, 2, 6, 19, 50, 116, 252, 528, 1084, 2200, 4436, 8912, 17868, 35784, 71620, 143296, 286652, 573368, 1146804, 2293680, 4587436, 9174952, 18349988, 36700064, 73400220, 146800536, 293601172, 587202448, 1174405004, 2348810120, 4697620356, 9395240832, 18790481788, 37580963704, 75161927540, 150323855216, 300647710572, 601295421288, 1202590842724, 2405181685600, 4810363371356, 9620726742872, 19241453485908, 38482906971984, 76965813944140, 153931627888456, 307863255777092, 615726511554368, 1231453023108924, 2462906046218040, 4925812092436276, 9851624184872752, 19703248369745708, 39406496739491624, 78812993478983460, 157625986957967136, 315251973915934492, 630503947831869208, 1261007895663738644, 2522015791327477520, 5044031582654955276, 10088063165309910792, 20176126330619821828, 40352252661239643904, 80704505322479288060, 161409010644958576376, 322818021289917153012, 645636042579834306288, 1291272085159668612844, 2582544170319337225960, 5165088340638674452196, 10330176681277348904672, 20660353362554697809628, 41320706725109395619544, 82641413450218791239380, 165282826900437582479056, 330565653800875164958412, 661131307601750329917128, 1322262615203500659834564, 2644525230407001319669440, 5289050460814002639339196, 10578100921628005278678712, 21156201843256010557357748, 42312403686512021114715824, 84624807373024042229431980, 169249614746048084458864296, 338499229492096168917728932, 676998458984192337835458208, 1353996917968384675670916764, 2707993835936769351341833880, 5415987671873538702683668116, 10831975343747077405367336592, 21663950687494154810734673548, 43327901374988309621469347464, 86655802749976619242938695300, 173311605499953238485877390976, 346623210999906476971754782332, 693246421999812953943509565048, 1386492843999625907887019130484, 2772985687999251815774038261360
Generating function in Maple syntax:
(x^6-3*x^4-x^3-3*x^2+3*x-1)/(2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{6}-3 x^{4}-x^{3}-3 x^{2}+3 x -1}{\left(2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**6 - 3*x**4 - x**3 - 3*x**2 + 3*x - 1)/((x - 1)**2*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x-1)^2*F(x)+x^6-3*x^4-x^3-3*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{6}-3 x^{4}-x^{3}-3 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,35/16*2^n-4*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{35 \,2^{n}}{16}-4 n & \mathit{\text{otherwise}} \end{array}\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) = 116
a(n+1) = 2*a(n)+4*n-4, 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) = 50
a \! \left(6\right) = 116
a \! \left(n +1\right) = 2 a \! \left(n \right)+4 n -4, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19506/
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[12,x]*F[4,x]
F[4,x] = F[26,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[12,x]
F[17,x] = F[18,x]+F[19,x]+F[23,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[10,x]*F[12,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[12,x]+F[25,x]
F[25,x] = F[18,x]+F[22,x]+F[23,x]
F[26,x] = F[27,x]+F[42,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[16,x]
F[33,x] = F[27,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[1,x]+F[37,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[37,x]
F[42,x] = F[18,x]+F[43,x]+F[77,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[56,x]
F[45,x] = F[46,x]+F[6,x]
F[46,x] = F[18,x]+F[47,x]+F[49,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[10,x]+F[25,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[24,x]+F[51,x]
F[51,x] = F[52,x]+F[54,x]
F[52,x] = F[53,x]
F[53,x] = x^2
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[21,x]
F[56,x] = F[42,x]+F[57,x]
F[57,x] = 2*F[18,x]+F[58,x]+F[69,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[65,x]
F[60,x] = F[18,x]+F[61,x]+F[63,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[10,x]+F[60,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[37,x]+F[60,x]
F[65,x] = 2*F[18,x]+F[66,x]+F[67,x]
F[66,x] = F[12,x]*F[60,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[40,x]+F[65,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[68,x]+F[71,x]
F[71,x] = F[72,x]+F[74,x]
F[72,x] = F[73,x]
F[73,x] = F[12,x]*F[40,x]
F[74,x] = F[75,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[66,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[64,x]+F[79,x]
F[79,x] = F[80,x]+F[84,x]
F[80,x] = F[18,x]+F[81,x]+F[83,x]
F[81,x] = F[12,x]*F[82,x]
F[82,x] = F[12,x]+F[40,x]
F[83,x] = F[12,x]*F[36,x]
F[84,x] = F[18,x]+F[67,x]+F[85,x]+F[87,x]
F[85,x] = F[12,x]*F[86,x]
F[86,x] = F[25,x]+F[65,x]
F[87,x] = F[12,x]*F[88,x]
F[88,x] = F[60,x]+F[76,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{26}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \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_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{42}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)+F_{43}\! \left(x \right)+F_{77}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{56}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{6}\! \left(x \right)
F_{46}\! \left(x \right) = F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{24}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = x^{2}
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{56}\! \left(x \right) = F_{42}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{58}\! \left(x \right)+F_{69}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{65}\! \left(x \right)
F_{60}\! \left(x \right) = F_{18}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{10}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{37}\! \left(x \right)+F_{60}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{40}\! \left(x \right)+F_{65}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{74}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{66}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{64}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{84}\! \left(x \right)
F_{80}\! \left(x \right) = F_{18}\! \left(x \right)+F_{81}\! \left(x \right)+F_{83}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right)+F_{40}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{84}\! \left(x \right) = F_{18}\! \left(x \right)+F_{67}\! \left(x \right)+F_{85}\! \left(x \right)+F_{87}\! \left(x \right)
F_{85}\! \left(x \right) = F_{12}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{25}\! \left(x \right)+F_{65}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{60}\! \left(x \right)+F_{76}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_26(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_12(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_23(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_10(x)*F_12(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_12(x) + F_25(x))
Eq(F_25(x), F_18(x) + F_22(x) + F_23(x))
Eq(F_26(x), F_27(x) + F_42(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_16(x))
Eq(F_33(x), F_27(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_1(x) + F_37(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_37(x))
Eq(F_42(x), F_18(x) + F_43(x) + F_77(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_56(x))
Eq(F_45(x), F_46(x) + F_6(x))
Eq(F_46(x), F_18(x) + F_47(x) + F_49(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_10(x) + F_25(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_24(x) + F_51(x))
Eq(F_51(x), F_52(x) + F_54(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), x**2)
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_21(x))
Eq(F_56(x), F_42(x) + F_57(x))
Eq(F_57(x), 2*F_18(x) + F_58(x) + F_69(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_65(x))
Eq(F_60(x), F_18(x) + F_61(x) + F_63(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_10(x) + F_60(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_37(x) + F_60(x))
Eq(F_65(x), 2*F_18(x) + F_66(x) + F_67(x))
Eq(F_66(x), F_12(x)*F_60(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_40(x) + F_65(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_68(x) + F_71(x))
Eq(F_71(x), F_72(x) + F_74(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_12(x)*F_40(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_66(x))
Eq(F_77(x), F_12(x)*F_78(x))
Eq(F_78(x), F_64(x) + F_79(x))
Eq(F_79(x), F_80(x) + F_84(x))
Eq(F_80(x), F_18(x) + F_81(x) + F_83(x))
Eq(F_81(x), F_12(x)*F_82(x))
Eq(F_82(x), F_12(x) + F_40(x))
Eq(F_83(x), F_12(x)*F_36(x))
Eq(F_84(x), F_18(x) + F_67(x) + F_85(x) + F_87(x))
Eq(F_85(x), F_12(x)*F_86(x))
Eq(F_86(x), F_25(x) + F_65(x))
Eq(F_87(x), F_12(x)*F_88(x))
Eq(F_88(x), F_60(x) + F_76(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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