0132_0231_1230_1302_2310
Counting sequence:
1, 1, 2, 6, 19, 55, 145, 356, 831, 1873, 4121, 8918, 19079, 40491, 85441, 179528, 375999, 785445, 1637241, 3406522, 7076327, 14678271, 30406657, 62912236, 130020799, 268432505, 553644825, 1140847006, 2348806151, 4831833683, 9932106881, 20401089168, 41875925119, 85899339341, 176093651961, 360777245058, 738734366439, 1511828479015, 3092376443201, 6322191849012, 12919261614847, 26388279054241, 53876069747737, 109951162763366, 224300372051079, 457396837138555, 932385860336705, 1899956092778456, 3870280929751871, 7881299347877493, 16044073672485241, 32651097298412618, 66428094503689959, 135107988821088591, 274719577269572481, 558446353793912188, 1134907106097334079, 2305843009213661385, 4683743612465281561, 9511602413006451502, 19311435202164648967, 39199331156632757411, 79551583817872399617, 161409010644958532896, 327429707308344495359, 664082786653543810205, 1346612317380797217785, 2730118122909013586706, 5534023222112865429991, 11215620396815407325303, 22726388698810167531201, 46043073207979040771268, 93266738036675492905599, 188894659314785808480241, 382511685112441262239001, 774468103190621814972918, 1567825672312722210870919, 3173430276488401583524555, 6422418416702717490544321, 12995952560857263628005992, 26294136576618184549770687, 53192736063043683686979781, 107594397945701996548754297, 217606647530633251447012826, 440048998339725019592945639, 889769403236367072583639455, 1798881619586568211962680065, 3636448865400804557516063756, 7350268983256945382213432511, 14855280471424563298789369113, 30020045952670471666303636761, 60659061924983633470056957118, 122556063889252647215013164039, 247588007857076054979824706291, 500127775871293631059246043521, 1010159072056870304317685219248, 2040125184742306693033756568959, 4119864450741745554864285260525, 8318957063997755447322114623481, 16796370453024039569831317304482, 33909653556105136490036810572007
Generating function in Maple syntax:
(x^6-8*x^5+20*x^4-28*x^3+20*x^2-7*x+1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
\frac{x^{6}-8 x^{5}+20 x^{4}-28 x^{3}+20 x^{2}-7 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**6 - 8*x**5 + 20*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^4*F(x)-x^6+8*x^5-20*x^4+28*x^3-20*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)-x^{6}+8 x^{5}-20 x^{4}+28 x^{3}-20 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-4/3*n-1-1/2*n^2-1/6*n^3+1/4*2^n*n+7/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -\frac{4 n}{3}-1-\frac{n^{2}}{2}-\frac{n^{3}}{6}+\frac{2^{n} n}{4}+\frac{7 \,2^{n}}{4} & \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) = 55
a(6) = 145
a(n+2) = -1/6*n^3+1/2*n^2-4*a(n)+4*a(n+1)+2/3*n+1, 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) = 55
a \! \left(6\right) = 145
a \! \left(n +2\right) = -\frac{n^{3}}{6}+\frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{2 n}{3}+1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19022/
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[22,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[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[33,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[16,x]+F[26,x]
F[26,x] = F[14,x]+F[27,x]
F[27,x] = F[18,x]+F[28,x]+F[30,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[14,x]+F[27,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[14,x]+F[32,x]
F[32,x] = F[30,x]
F[33,x] = F[18,x]+F[34,x]+F[64,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[41,x]
F[36,x] = F[37,x]+F[6,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[10,x]+F[40,x]
F[40,x] = F[38,x]
F[41,x] = F[42,x]+F[61,x]
F[42,x] = F[18,x]+F[43,x]+F[45,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[42,x]+F[6,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[13,x]+F[47,x]
F[47,x] = F[27,x]+F[48,x]
F[48,x] = F[18,x]+F[49,x]+F[55,x]+F[60,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[40,x]+F[51,x]
F[51,x] = 2*F[18,x]+F[49,x]+F[52,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[17,x]+F[54,x]
F[54,x] = F[52,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[17,x]+F[57,x]
F[57,x] = 2*F[18,x]+F[55,x]+F[58,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[32,x]+F[57,x]
F[60,x] = F[12,x]*F[47,x]
F[61,x] = 2*F[18,x]+F[52,x]+F[62,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[37,x]+F[61,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[66,x]+F[72,x]
F[66,x] = F[23,x]+F[67,x]
F[67,x] = F[18,x]+F[68,x]+F[71,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[39,x]+F[70,x]
F[70,x] = F[17,x]+F[51,x]
F[71,x] = F[12,x]*F[66,x]
F[72,x] = F[73,x]+F[78,x]
F[73,x] = F[18,x]+F[30,x]+F[74,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[76,x]+F[77,x]
F[76,x] = F[14,x]
F[77,x] = F[27,x]
F[78,x] = F[18,x]+F[55,x]+F[79,x]+F[83,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[81,x]+F[82,x]
F[81,x] = F[40,x]
F[82,x] = F[51,x]
F[83,x] = F[12,x]*F[72,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_{22}\! \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_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \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_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{33}\! \left(x \right)
F_{23}\! \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_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{64}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{6}\! \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_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{61}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{42}\! \left(x \right)+F_{6}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{27}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{55}\! \left(x \right)+F_{60}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{40}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{17}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{17}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{32}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{37}\! \left(x \right)+F_{61}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{72}\! \left(x \right)
F_{66}\! \left(x \right) = F_{23}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{18}\! \left(x \right)+F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{39}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{17}\! \left(x \right)+F_{51}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{78}\! \left(x \right)
F_{73}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{76}\! \left(x \right) = F_{14}\! \left(x \right)
F_{77}\! \left(x \right) = F_{27}\! \left(x \right)
F_{78}\! \left(x \right) = F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{79}\! \left(x \right)+F_{83}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{82}\! \left(x \right)
F_{81}\! \left(x \right) = F_{40}\! \left(x \right)
F_{82}\! \left(x \right) = F_{51}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \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_22(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_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_33(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_16(x) + F_26(x))
Eq(F_26(x), F_14(x) + F_27(x))
Eq(F_27(x), F_18(x) + F_28(x) + F_30(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_14(x) + F_27(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_14(x) + F_32(x))
Eq(F_32(x), F_30(x))
Eq(F_33(x), F_18(x) + F_34(x) + F_64(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_41(x))
Eq(F_36(x), F_37(x) + F_6(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_10(x) + F_40(x))
Eq(F_40(x), F_38(x))
Eq(F_41(x), F_42(x) + F_61(x))
Eq(F_42(x), F_18(x) + F_43(x) + F_45(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_42(x) + F_6(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_13(x) + F_47(x))
Eq(F_47(x), F_27(x) + F_48(x))
Eq(F_48(x), F_18(x) + F_49(x) + F_55(x) + F_60(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_40(x) + F_51(x))
Eq(F_51(x), 2*F_18(x) + F_49(x) + F_52(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_17(x) + F_54(x))
Eq(F_54(x), F_52(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_17(x) + F_57(x))
Eq(F_57(x), 2*F_18(x) + F_55(x) + F_58(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_32(x) + F_57(x))
Eq(F_60(x), F_12(x)*F_47(x))
Eq(F_61(x), 2*F_18(x) + F_52(x) + F_62(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_37(x) + F_61(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_72(x))
Eq(F_66(x), F_23(x) + F_67(x))
Eq(F_67(x), F_18(x) + F_68(x) + F_71(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_39(x) + F_70(x))
Eq(F_70(x), F_17(x) + F_51(x))
Eq(F_71(x), F_12(x)*F_66(x))
Eq(F_72(x), F_73(x) + F_78(x))
Eq(F_73(x), F_18(x) + F_30(x) + F_74(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_77(x))
Eq(F_76(x), F_14(x))
Eq(F_77(x), F_27(x))
Eq(F_78(x), F_18(x) + F_55(x) + F_79(x) + F_83(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_81(x) + F_82(x))
Eq(F_81(x), F_40(x))
Eq(F_82(x), F_51(x))
Eq(F_83(x), F_12(x)*F_72(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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