0132_0321_1230_1320_2130
Counting sequence:
1, 1, 2, 6, 19, 56, 155, 416, 1101, 2896, 7597, 19906, 52133, 136506, 357399, 935706, 2449735, 6413516, 16790831, 43958996, 115086177, 301299556, 788812513, 2065138006, 5406601529, 14154666606, 37057398315, 97017528366, 253995186811, 664968032096, 1740908909507, 4557758696456, 11932367179893, 31239342843256, 81785661349909, 214117641206506, 560567262269645, 1467584145602466, 3842185174537791, 10058971378010946, 26334728959495087, 68945215500474356, 180500917541928023, 472557537125309756, 1237171693834001289, 3238957544376694156, 8479700939296081225, 22200145273511549566, 58120734881238567521, 152162059370204153046, 398365443229373891667, 1042934270317917522006, 2730437367724378674403, 7148377832855218501256, 18714696130841276829419, 48995710559668611987056, 128272435548164559131805, 335821596084825065408416, 879192352706310637093501, 2301755462034106845872146, 6026074033396009900522997, 15776466638153922855696906, 41303325881065758666567783, 108133511005043353144006506, 283097207134064300765451799, 741158110397149549152348956, 1940377124057384346691595135, 5079973261775003490922436516, 13299542661267626126075714481, 34818654722027874887304706996, 91156421504815998535838406577, 238650609792420120720210512806, 624795407872444363624793131913, 1635735613824912970154168883006, 4282411433602294546837713517179, 11211498686981970670358971668606, 29352084627343617464239201488715, 76844755195048881722358632797616, 201182180957803027702836696904211, 526701787678360201386151457915096, 1378923182077277576455617676841157, 3610067758553472527980701572608456, 9451280093583140007486487040984293, 24743772522195947494478759550344506, 64780037473004702475949791610049309, 169596339896818159933370615279803506, 444008982217449777324162054229361295, 1162430606755531172039115547408280466, 3043282838049143738793184587995480191, 7967417907391900044340438216578160196, 20858970884126556394228130061739000487, 54609494744987769138343951968638841356, 142969513350836751020803725844177523673, 374299045307522483924067225563893729756, 979927622571730700751397950847503665689, 2565483822407669618330126626978617267406, 6716523844651278154238981930088348136625, 17584087711546164844386819163286427142566, 46035739289987216378921475559770933291171, 120523130158415484292377607516026372731046, 315533651185259236498211346988308184902067
Generating function in Maple syntax:
-(x^6-x^4+x^3-5*x^2+4*x-1)/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
-\frac{x^{6}-x^{4}+x^{3}-5 x^{2}+4 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**6 + x**4 - x**3 + 5*x**2 - 4*x + 1)/((x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^2*F(x)+x^6-x^4+x^3-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{6}-x^{4}+x^{3}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/5*(-19*5^(1/2)+45)*(3/2-1/2*5^(1/2))^(-n)+1/5*(19*5^(1/2)+45)*(3/2+1/2*5^(1/2))^(-n)-n-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-19 \sqrt{5}+45\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{5}+\frac{\left(19 \sqrt{5}+45\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{5}-n -1 & \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) = 56
a(6) = 155
a(n+2) = -a(n)+3*a(n+1)+n+2, 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) = 56
a \! \left(6\right) = 155
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+n +2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19484/
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[24,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[13,x]
F[24,x] = F[25,x]+F[31,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[16,x]+F[28,x]
F[28,x] = F[25,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[25,x]
F[31,x] = F[18,x]+F[32,x]+F[69,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[47,x]
F[34,x] = F[35,x]+F[6,x]
F[35,x] = F[18,x]+F[22,x]+F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[12,x]+F[39,x]
F[39,x] = F[18,x]+F[22,x]+F[40,x]
F[40,x] = F[12,x]*F[38,x]
F[41,x] = F[42,x]+F[44,x]
F[42,x] = F[43,x]
F[43,x] = x^2
F[44,x] = 2*F[18,x]+F[45,x]+F[46,x]
F[45,x] = F[12,x]*F[21,x]
F[46,x] = F[12,x]*F[41,x]
F[47,x] = F[31,x]+F[48,x]
F[48,x] = 2*F[18,x]+F[49,x]+F[60,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[18,x]+F[51,x]+F[58,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[10,x]+F[39,x]
F[54,x] = F[50,x]+F[55,x]
F[55,x] = 2*F[18,x]+F[49,x]+F[56,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[29,x]+F[55,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[25,x]+F[50,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[57,x]+F[62,x]
F[62,x] = F[63,x]+F[65,x]
F[63,x] = F[64,x]
F[64,x] = F[12,x]*F[29,x]
F[65,x] = 3*F[18,x]+F[66,x]+F[68,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[49,x]
F[68,x] = F[12,x]*F[62,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[59,x]+F[71,x]
F[71,x] = F[72,x]+F[78,x]
F[72,x] = F[18,x]+F[73,x]+F[77,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[75,x]+F[76,x]
F[75,x] = F[12,x]
F[76,x] = F[29,x]
F[77,x] = F[12,x]*F[28,x]
F[78,x] = F[18,x]+F[79,x]+F[83,x]+F[85,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[81,x]+F[82,x]
F[81,x] = F[39,x]
F[82,x] = F[55,x]
F[83,x] = F[12,x]*F[84,x]
F[84,x] = F[50,x]+F[67,x]
F[85,x] = F[12,x]*F[71,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_{24}\! \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_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{31}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{16}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{31}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)+F_{69}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{47}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{6}\! \left(x \right)
F_{35}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = x^{2}
F_{44}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{47}\! \left(x \right) = F_{31}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{49}\! \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_{18}\! \left(x \right)+F_{51}\! \left(x \right)+F_{58}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{10}\! \left(x \right)+F_{39}\! \left(x \right)
F_{54}\! \left(x \right) = F_{50}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{29}\! \left(x \right)+F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{25}\! \left(x \right)+F_{50}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{57}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{65}\! \left(x \right) = 3 F_{18}\! \left(x \right)+F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{49}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{59}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{78}\! \left(x \right)
F_{72}\! \left(x \right) = F_{18}\! \left(x \right)+F_{73}\! \left(x \right)+F_{77}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{76}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right)
F_{76}\! \left(x \right) = F_{29}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{78}\! \left(x \right) = F_{18}\! \left(x \right)+F_{79}\! \left(x \right)+F_{83}\! \left(x \right)+F_{85}\! \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_{39}\! \left(x \right)
F_{82}\! \left(x \right) = F_{55}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{50}\! \left(x \right)+F_{67}\! \left(x \right)
F_{85}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \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_24(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_13(x))
Eq(F_24(x), F_25(x) + F_31(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_16(x) + F_28(x))
Eq(F_28(x), F_25(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_25(x))
Eq(F_31(x), F_18(x) + F_32(x) + F_69(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_47(x))
Eq(F_34(x), F_35(x) + F_6(x))
Eq(F_35(x), F_18(x) + F_22(x) + F_36(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_12(x) + F_39(x))
Eq(F_39(x), F_18(x) + F_22(x) + F_40(x))
Eq(F_40(x), F_12(x)*F_38(x))
Eq(F_41(x), F_42(x) + F_44(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), x**2)
Eq(F_44(x), 2*F_18(x) + F_45(x) + F_46(x))
Eq(F_45(x), F_12(x)*F_21(x))
Eq(F_46(x), F_12(x)*F_41(x))
Eq(F_47(x), F_31(x) + F_48(x))
Eq(F_48(x), 2*F_18(x) + F_49(x) + F_60(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_18(x) + F_51(x) + F_58(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_10(x) + F_39(x))
Eq(F_54(x), F_50(x) + F_55(x))
Eq(F_55(x), 2*F_18(x) + F_49(x) + F_56(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_29(x) + F_55(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_25(x) + F_50(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_57(x) + F_62(x))
Eq(F_62(x), F_63(x) + F_65(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_12(x)*F_29(x))
Eq(F_65(x), 3*F_18(x) + F_66(x) + F_68(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_49(x))
Eq(F_68(x), F_12(x)*F_62(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_59(x) + F_71(x))
Eq(F_71(x), F_72(x) + F_78(x))
Eq(F_72(x), F_18(x) + F_73(x) + F_77(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_76(x))
Eq(F_75(x), F_12(x))
Eq(F_76(x), F_29(x))
Eq(F_77(x), F_12(x)*F_28(x))
Eq(F_78(x), F_18(x) + F_79(x) + F_83(x) + F_85(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_39(x))
Eq(F_82(x), F_55(x))
Eq(F_83(x), F_12(x)*F_84(x))
Eq(F_84(x), F_50(x) + F_67(x))
Eq(F_85(x), F_12(x)*F_71(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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