0132_1203_2130_3102
Counting sequence:
1, 1, 2, 6, 20, 60, 163, 420, 1058, 2646, 6613, 16553, 41521, 104382, 263026, 664448, 1683037, 4275251, 10891785, 27829448, 71308834, 183211586, 471901173, 1218268021, 3151573665, 8167767770, 21201779298, 55111458460, 143425850733, 373639361311, 974203892393, 2541909100708, 6636390352866, 17334915476542, 45299368154741, 118416354782433, 309637774180817, 809835542783446, 2118468694816850, 5542611309184728, 14503120350272189, 37953589961818731, 99329955586039753, 259978071399483936, 680482075382752610, 1781211968539477530, 4662617268561298933, 12205517235772431725, 31951588080942292033, 83644347583241042226, 218971233608688994498, 573248049105526116788, 1500728546214685299661, 3928845265746168645527, 10285615285089703925481, 26927601739480944084668, 70496361833971645529122, 184559765639176150606166, 483179372736251556171765, 1264970971169789257607833, 3311718255555980921126769, 8670152162214370855343726, 22698672802791745206270258, 59425730994056855572659376, 155578240740029142427096605, 407308414188817515981639715, 1066345810858509146644813193, 2791726561510983606761806840, 7308828807737350431154903458, 19134749420843921967096593266, 50095397945890825524797592053, 131151400126196033630662648293, 343358711268888063094586157281, 898924546113974414174374877962, 2353414541314723110014394963746, 6161318284767773698947766814028, 16130538683181865586442600972077, 42230294416619986705801579456911, 110560337690912495472758706116905, 289450704540964082415104995010388, 757791746964983745850152537951074, 1983924476927727334703737249534446, 5193981561943364068361255535421813, 13598019959032509528781361809477841, 35600077803022964183735545757931025, 93202212400688771596927151819170950, 244006557249573243332235237071810578, 638817454946332049182913734290876680, 1672445798578089844809060097540779133, 4378519922344565181203483869013827931, 11463113930717777313711127191872495433, 30010821792611599622618137126565856848, 78569351289240980919029631004595904226, 205697231752318477890807849207274971722, 538522343307892411814286230095995548661, 1409869796822932569760450376646840354013, 3691087044405869740037451098056524285057, 9663391330767001231760564187831313502306, 25299086936402098630518770993273946640770, 66233869454972903275102080301002727335268, 173402521380613283434600394792237724089229
Generating function in Maple syntax:
-(3*x^11-18*x^10+26*x^9+31*x^8-138*x^7+263*x^6-352*x^5+315*x^4-179*x^3+62*x^2-12*x+1)/(x^2-3*x+1)/(2*x-1)^3/(x-1)^4
Generating function in latex syntax:
-\frac{3 x^{11}-18 x^{10}+26 x^{9}+31 x^{8}-138 x^{7}+263 x^{6}-352 x^{5}+315 x^{4}-179 x^{3}+62 x^{2}-12 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{3} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-3*x**11 + 18*x**10 - 26*x**9 - 31*x**8 + 138*x**7 - 263*x**6 + 352*x**5 - 315*x**4 + 179*x**3 - 62*x**2 + 12*x - 1)/((x - 1)**4*(2*x - 1)**3*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)^3*(x-1)^4*F(x)+3*x^11-18*x^10+26*x^9+31*x^8-138*x^7+263*x^6-352*x^5+315*x^4-179*x^3+62*x^2-12*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{3} \left(x -1\right)^{4} F \! \left(x \right)+3 x^{11}-18 x^{10}+26 x^{9}+31 x^{8}-138 x^{7}+263 x^{6}-352 x^{5}+315 x^{4}-179 x^{3}+62 x^{2}-12 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/960*(-96*5^(1/2)+480)*(3/2-1/2*5^(1/2))^(-n)+1/960*(96*5^(1/2)+480)*(3/2+1/2*5^(1/2))^(-n)+1/960*(15*n^2+75*n-120)*2^n+1/3*n^3-2*n^2+8/3*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-96 \sqrt{5}+480\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{960}+\frac{\left(96 \sqrt{5}+480\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{960}+\\\frac{\left(15 n^{2}+75 n -120\right) 2^{n}}{960}+\frac{n^{3}}{3}-2 n^{2}+\frac{8 n}{3} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 60
a(6) = 163
a(7) = 420
a(8) = 1058
a(9) = 2646
a(10) = 6613
a(11) = 16553
a(n+5) = 1/3*n^3-4*n^2+8*a(n)-36*a(n+1)+50*a(n+2)-31*a(n+3)+9*a(n+4)+20/3*n+10, n >= 12
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) = 20
a \! \left(5\right) = 60
a \! \left(6\right) = 163
a \! \left(7\right) = 420
a \! \left(8\right) = 1058
a \! \left(9\right) = 2646
a \! \left(10\right) = 6613
a \! \left(11\right) = 16553
a \! \left(n +5\right) = \frac{n^{3}}{3}-4 n^{2}+8 a \! \left(n \right)-36 a \! \left(n +1\right)+50 a \! \left(n +2\right)-31 a \! \left(n +3\right)+9 a \! \left(n +4\right)+\frac{20 n}{3}+10, \quad n \geq 12
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15594/
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[28,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[20,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[19,x]+F[21,x]+F[22,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[24,x]*F[4,x]
F[27,x] = F[22,x]
F[28,x] = F[29,x]+F[43,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[1,x]+F[14,x]
F[33,x] = F[29,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[29,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[37,x]*F[4,x]
F[40,x] = F[34,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[21,x]+F[44,x]+F[66,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[60,x]
F[46,x] = F[47,x]+F[7,x]
F[47,x] = F[21,x]+F[22,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[17,x]+F[50,x]
F[50,x] = F[51,x]+F[59,x]
F[51,x] = F[21,x]+F[52,x]+F[54,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[24,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[17,x]+F[56,x]
F[56,x] = F[51,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[56,x]
F[59,x] = F[58,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[65,x]
F[63,x] = F[64,x]+F[7,x]
F[64,x] = F[22,x]
F[65,x] = F[61,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[37,x]+F[68,x]
F[68,x] = F[69,x]+F[78,x]
F[69,x] = F[21,x]+F[70,x]+F[73,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[14,x]+F[51,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[37,x]+F[75,x]
F[75,x] = F[69,x]+F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[4,x]*F[75,x]
F[78,x] = F[77,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_{28}\! \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_{20}\! \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_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{19}\! \left(x \right)+F_{21}\! \left(x \right)+F_{22}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{43}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{33}\! \left(x \right) = F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{29}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{21}\! \left(x \right)+F_{44}\! \left(x \right)+F_{66}\! \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_{60}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{7}\! \left(x \right)
F_{47}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{17}\! \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_{21}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{24}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{17}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{51}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{58}\! \left(x \right)
F_{60}\! \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_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{7}\! \left(x \right)
F_{64}\! \left(x \right) = F_{22}\! \left(x \right)
F_{65}\! \left(x \right) = F_{61}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{37}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{78}\! \left(x \right)
F_{69}\! \left(x \right) = F_{21}\! \left(x \right)+F_{70}\! \left(x \right)+F_{73}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{14}\! \left(x \right)+F_{51}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{37}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{69}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{78}\! \left(x \right) = F_{77}\! \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_28(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_20(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_19(x) + F_21(x) + F_22(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_24(x)*F_4(x))
Eq(F_27(x), F_22(x))
Eq(F_28(x), F_29(x) + F_43(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_1(x) + F_14(x))
Eq(F_33(x), F_29(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_29(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_37(x)*F_4(x))
Eq(F_40(x), F_34(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), F_21(x) + F_44(x) + F_66(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_60(x))
Eq(F_46(x), F_47(x) + F_7(x))
Eq(F_47(x), F_21(x) + F_22(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_17(x) + F_50(x))
Eq(F_50(x), F_51(x) + F_59(x))
Eq(F_51(x), F_21(x) + F_52(x) + F_54(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_24(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_17(x) + F_56(x))
Eq(F_56(x), F_51(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_56(x))
Eq(F_59(x), F_58(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_65(x))
Eq(F_63(x), F_64(x) + F_7(x))
Eq(F_64(x), F_22(x))
Eq(F_65(x), F_61(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_37(x) + F_68(x))
Eq(F_68(x), F_69(x) + F_78(x))
Eq(F_69(x), F_21(x) + F_70(x) + F_73(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_14(x) + F_51(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_37(x) + F_75(x))
Eq(F_75(x), F_69(x) + F_76(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_4(x)*F_75(x))
Eq(F_78(x), F_77(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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