0132_1203_2031_3120

Counting sequence:
1, 1, 2, 6, 20, 63, 184, 505, 1321, 3328, 8135, 19400, 45326, 104101, 235674, 527099, 1166807, 2560318, 5576001, 12065266, 25960292, 55584171, 118500572, 251671821, 532692565, 1124092108, 2365609099, 4966080940, 10401902666, 21743304833, 45365629550, 94489323055, 196494801891, 408021947306, 846108618149, 1752346724854, 3624952473792, 7490423048743, 15461882359376, 31885837309281, 65695819874177, 135239930342616, 278176441965967, 571746046595536, 1174278418631110, 2410129488266653, 4943404278678914, 10133099161799075, 20758779532644911, 42502721483562838, 86975767803867465, 177892185281431610, 363665669910488156, 743093938516477411, 1517713074424229124, 3098476543631301109, 6323053876828605677, 12898309332789560868, 26301021823844189651, 53610849964218911796, 109239312561499314562, 222513850389122067961, 453098151310491503062, 922337203685478264151, 1876956209499947604155, 3818476023257877957698, 7766079255031722051629, 15790412927095377054910, 32097334688254620736056, 65227687044636975493343, 132521409425529419845880, 269174889523569778277065, 546613920392161434644121, 1109756123474366626442416, 2252568812328820768224599, 4571250755417816568219864, 9274727772355983201134462, 18813908067752666532876693, 38156721181586733328254826, 77371252455336267182868811, 156858125094998135419885447, 317947490558647472949572206, 644357461854597350120331857, 1305639885183799508684705666, 2645129693316808634259247700, 5357959232532036502300009243, 10851318156860911472164979244, 21973435697315499879461908445, 44488470161818353629189844101, 90060137858011414998913972348, 182286670784772245478898848795, 368906131707043321919941951420, 746477843689084305764174969786, 1510286847928163935376934750321, 3055236016956318518451041920574, 6179796676112618332296431605119, 12498242636625199255381561792147, 25273783842050323692340523936218, 51102164821700497747835851902901, 103313523918600696221981315336326, 208845436387600793896581857350832

Generating function in Maple syntax:
(x^7-10*x^6+29*x^5-49*x^4+48*x^3-27*x^2+8*x-1)/(2*x-1)^2/(x-1)^5

Generating function in latex syntax:
\frac{x^{7}-10 x^{6}+29 x^{5}-49 x^{4}+48 x^{3}-27 x^{2}+8 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}

Generating function in sympy syntax:
(x**7 - 10*x**6 + 29*x**5 - 49*x**4 + 48*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**5*(2*x - 1)**2)

Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^5*F(x)-x^7+10*x^6-29*x^5+49*x^4-48*x^3+27*x^2-8*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)-x^{7}+10 x^{6}-29 x^{5}+49 x^{4}-48 x^{3}+27 x^{2}-8 x +1 = 0

Explicit closed form in Maple syntax:
piecewise(n = 0,1,11+7/4*2^n*n-41/4*2^n+35/24*n^2+65/12*n+1/12*n^3+1/24*n^4)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 11+\frac{7 \,2^{n} n}{4}-\frac{41 \,2^{n}}{4}+\frac{35 n^{2}}{24}+\frac{65 n}{12}+\frac{n^{3}}{12}+\frac{n^{4}}{24} & \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) = 63
a(6) = 184
a(7) = 505
a(n+2) = 1/24*n^4-1/4*n^3+23/24*n^2-4*a(n)+4*a(n+1)+1/4*n+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) = 20
a \! \left(5\right) = 63
a \! \left(6\right) = 184
a \! \left(7\right) = 505
a \! \left(n +2\right) = \frac{n^{4}}{24}-\frac{n^{3}}{4}+\frac{23 n^{2}}{24}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{n}{4}+1, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15581/
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[29,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[23,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[22,x]
F[21,x] = F[10,x]
F[22,x] = F[18,x]
F[23,x] = F[24,x]+F[25,x]+F[28,x]
F[24,x] = 0
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[21,x]+F[27,x]
F[27,x] = F[23,x]
F[28,x] = F[12,x]*F[13,x]
F[29,x] = F[30,x]+F[40,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[1,x]+F[14,x]
F[34,x] = F[30,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[14,x]
F[39,x] = F[35,x]
F[40,x] = F[24,x]+F[41,x]+F[68,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[43,x]+F[62,x]
F[43,x] = F[44,x]+F[6,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[14,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[47,x]
F[50,x] = F[51,x]+F[56,x]
F[51,x] = F[52,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[14,x]
F[55,x] = F[51,x]
F[56,x] = 2*F[24,x]+F[57,x]+F[61,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[48,x]
F[60,x] = F[56,x]
F[61,x] = F[12,x]*F[50,x]
F[62,x] = F[40,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[44,x]
F[67,x] = F[63,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[70,x]+F[82,x]
F[70,x] = F[30,x]+F[71,x]
F[71,x] = F[24,x]+F[72,x]+F[81,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[10,x]+F[48,x]
F[75,x] = F[71,x]+F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[79,x]+F[80,x]
F[79,x] = F[48,x]
F[80,x] = F[76,x]
F[81,x] = F[12,x]*F[70,x]
F[82,x] = F[83,x]+F[88,x]
F[83,x] = F[84,x]
F[84,x] = F[12,x]*F[85,x]
F[85,x] = F[86,x]+F[87,x]
F[86,x] = F[30,x]
F[87,x] = F[83,x]
F[88,x] = 2*F[24,x]+F[89,x]+F[93,x]
F[89,x] = F[12,x]*F[90,x]
F[90,x] = F[91,x]+F[92,x]
F[91,x] = F[71,x]
F[92,x] = F[88,x]
F[93,x] = F[12,x]*F[82,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_{29}\! \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_{23}\! \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_{17}\! \left(x \right)+F_{9}\! \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_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{40}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \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_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{24}\! \left(x \right)+F_{41}\! \left(x \right)+F_{68}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{62}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{6}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{14}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{51}\! \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_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{14}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{48}\! \left(x \right)
F_{60}\! \left(x \right) = F_{56}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{62}\! \left(x \right) = F_{40}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \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_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{44}\! \left(x \right)
F_{67}\! \left(x \right) = F_{63}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{82}\! \left(x \right)
F_{70}\! \left(x \right) = F_{30}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{24}\! \left(x \right)+F_{72}\! \left(x \right)+F_{81}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{10}\! \left(x \right)+F_{48}\! \left(x \right)
F_{75}\! \left(x \right) = F_{71}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{80}\! \left(x \right)
F_{79}\! \left(x \right) = F_{48}\! \left(x \right)
F_{80}\! \left(x \right) = F_{76}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{88}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)+F_{87}\! \left(x \right)
F_{86}\! \left(x \right) = F_{30}\! \left(x \right)
F_{87}\! \left(x \right) = F_{83}\! \left(x \right)
F_{88}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{89}\! \left(x \right)+F_{93}\! \left(x \right)
F_{89}\! \left(x \right) = F_{12}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)+F_{92}\! \left(x \right)
F_{91}\! \left(x \right) = F_{71}\! \left(x \right)
F_{92}\! \left(x \right) = F_{88}\! \left(x \right)
F_{93}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \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_29(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_23(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_22(x))
Eq(F_21(x), F_10(x))
Eq(F_22(x), F_18(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_28(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_21(x) + F_27(x))
Eq(F_27(x), F_23(x))
Eq(F_28(x), F_12(x)*F_13(x))
Eq(F_29(x), F_30(x) + F_40(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_1(x) + F_14(x))
Eq(F_34(x), F_30(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_14(x))
Eq(F_39(x), F_35(x))
Eq(F_40(x), F_24(x) + F_41(x) + F_68(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_62(x))
Eq(F_43(x), F_44(x) + F_6(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_14(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_47(x))
Eq(F_50(x), F_51(x) + F_56(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_14(x))
Eq(F_55(x), F_51(x))
Eq(F_56(x), 2*F_24(x) + F_57(x) + F_61(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_48(x))
Eq(F_60(x), F_56(x))
Eq(F_61(x), F_12(x)*F_50(x))
Eq(F_62(x), F_40(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_67(x))
Eq(F_66(x), F_44(x))
Eq(F_67(x), F_63(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_82(x))
Eq(F_70(x), F_30(x) + F_71(x))
Eq(F_71(x), F_24(x) + F_72(x) + F_81(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_10(x) + F_48(x))
Eq(F_75(x), F_71(x) + F_76(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_12(x)*F_78(x))
Eq(F_78(x), F_79(x) + F_80(x))
Eq(F_79(x), F_48(x))
Eq(F_80(x), F_76(x))
Eq(F_81(x), F_12(x)*F_70(x))
Eq(F_82(x), F_83(x) + F_88(x))
Eq(F_83(x), F_84(x))
Eq(F_84(x), F_12(x)*F_85(x))
Eq(F_85(x), F_86(x) + F_87(x))
Eq(F_86(x), F_30(x))
Eq(F_87(x), F_83(x))
Eq(F_88(x), 2*F_24(x) + F_89(x) + F_93(x))
Eq(F_89(x), F_12(x)*F_90(x))
Eq(F_90(x), F_91(x) + F_92(x))
Eq(F_91(x), F_71(x))
Eq(F_92(x), F_88(x))
Eq(F_93(x), F_12(x)*F_82(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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