0132_0231_1203_1320_2130
Counting sequence:
1, 1, 2, 6, 19, 56, 155, 409, 1041, 2577, 6241, 14849, 34817, 80641, 184833, 419841, 946177, 2117633, 4710401, 10420225, 22937601, 50266113, 109707265, 238551041, 516947969, 1116733441, 2405433345, 5167382529, 11072962561, 23672651777, 50499420161, 107508400129, 228438573057, 484525998081, 1025960312833, 2168958484481, 4578435137537, 9650791514113, 20315195310081, 42709154791425, 89678917140481, 188085207826433, 394037479604225, 824633720832001, 1724034232352769, 3600900580966401, 7514062464221185, 15665841672552449, 32633505112391681, 67923430317490177, 141265253936660481, 293578400709214209, 609674799555280897, 1265230020314398721, 2623909732896735233, 5438096550049873921, 11263502668053610497, 23315135270897057793, 48233552009138012161, 99727710148492263425, 206084718948473896961, 425644207982040317953, 878670301698493251585, 1812969065994266869761, 3738924439440004743169, 7707280258296772034561, 15880340804454710247425, 32706077242687035015169, 67330615869039863398401, 138553494737632442187777, 285002195938812572467201, 586016165733605035737089, 1204498601036938882318337, 2474815184928873444802561, 5083037222998814366892033, 10436429927141915922268161, 21420654366296710689325057, 43951064856067788003868673, 90149976157981527129456641, 184852313605449392244916225, 378922686585460331946967041, 776508165511221501778460673, 1590795262885400165266489345, 3258055083861425625833144321, 6670852672633523786028679169, 13654817132547236528306257921, 27943111394572538744158552065, 57167684157936584413505650689, 116927305273126933777581342721, 239096512900102899656688664577, 488792887386586867917201080321, 999017611703301881843593248769, 2041363124781592073308655845377, 4170310507342624801066424729601, 8517646440303058981443424223233, 17393057551959592862332695347201, 35509072086861847805206479241217, 72478913420080444334793925066753, 147909075893817235244947362283521, 301780071016832861893808906829825, 615602822735833903101836494110721
Generating function in Maple syntax:
(x^6-x^5+x^4-10*x^3+13*x^2-6*x+1)/(x-1)/(2*x-1)^3
Generating function in latex syntax:
\frac{x^{6}-x^{5}+x^{4}-10 x^{3}+13 x^{2}-6 x +1}{\left(x -1\right) \left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(x**6 - x**5 + x**4 - 10*x**3 + 13*x**2 - 6*x + 1)/((x - 1)*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x-1)^3*F(x)+x^6-x^5+x^4-10*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x -1\right)^{3} F \! \left(x \right)+x^{6}-x^{5}+x^{4}-10 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1+1/64*(3*n^2+11*n-20)*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 1+\frac{\left(3 n^{2}+11 n -20\right) 2^{n}}{64} & \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+3) = 8*a(n)-12*a(n+1)+6*a(n+2)-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) = 56
a \! \left(6\right) = 155
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18959/
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[29,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[9,x]
F[26,x] = F[23,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[26,x]
F[29,x] = F[18,x]+F[30,x]+F[64,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[47,x]
F[32,x] = F[33,x]+F[6,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[12,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[36,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[12,x]+F[40,x]
F[43,x] = 2*F[18,x]+F[44,x]+F[46,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[37,x]+F[43,x]
F[46,x] = F[12,x]*F[39,x]
F[47,x] = F[29,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[56,x]
F[51,x] = F[52,x]+F[54,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[23,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[51,x]
F[56,x] = F[57,x]+F[60,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[52,x]+F[57,x]
F[60,x] = 3*F[18,x]+F[61,x]+F[63,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[54,x]+F[60,x]
F[63,x] = F[12,x]*F[56,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[66,x]+F[75,x]
F[66,x] = F[23,x]+F[67,x]
F[67,x] = F[18,x]+F[68,x]+F[74,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[70,x]+F[72,x]
F[70,x] = F[10,x]+F[71,x]
F[71,x] = F[38,x]
F[72,x] = F[67,x]+F[73,x]
F[73,x] = F[55,x]
F[74,x] = F[12,x]*F[66,x]
F[75,x] = F[76,x]+F[83,x]
F[76,x] = F[18,x]+F[77,x]+F[81,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[79,x]+F[80,x]
F[79,x] = F[12,x]
F[80,x] = F[52,x]
F[81,x] = F[12,x]*F[82,x]
F[82,x] = F[23,x]+F[76,x]
F[83,x] = F[18,x]+F[84,x]+F[88,x]+F[90,x]
F[84,x] = F[12,x]*F[85,x]
F[85,x] = F[86,x]+F[87,x]
F[86,x] = F[37,x]
F[87,x] = F[54,x]
F[88,x] = F[12,x]*F[89,x]
F[89,x] = F[67,x]+F[83,x]
F[90,x] = F[12,x]*F[75,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_{29}\! \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_{26}\! \left(x \right)+F_{9}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{64}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{47}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \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_{39}\! \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_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{37}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{47}\! \left(x \right) = F_{29}\! \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_{50}\! \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_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{60}\! \left(x \right)
F_{57}\! \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_{52}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = 3 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_{54}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \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_{75}\! \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_{74}\! \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_{72}\! \left(x \right)
F_{70}\! \left(x \right) = F_{10}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{38}\! \left(x \right)
F_{72}\! \left(x \right) = F_{67}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{55}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{83}\! \left(x \right)
F_{76}\! \left(x \right) = F_{18}\! \left(x \right)+F_{77}\! \left(x \right)+F_{81}\! \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_{12}\! \left(x \right)
F_{80}\! \left(x \right) = F_{52}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{23}\! \left(x \right)+F_{76}\! \left(x \right)
F_{83}\! \left(x \right) = F_{18}\! \left(x \right)+F_{84}\! \left(x \right)+F_{88}\! \left(x \right)+F_{90}\! \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_{37}\! \left(x \right)
F_{87}\! \left(x \right) = F_{54}\! \left(x \right)
F_{88}\! \left(x \right) = F_{12}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{67}\! \left(x \right)+F_{83}\! \left(x \right)
F_{90}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \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_29(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_9(x))
Eq(F_26(x), F_23(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_26(x))
Eq(F_29(x), F_18(x) + F_30(x) + F_64(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_47(x))
Eq(F_32(x), F_33(x) + F_6(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_12(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_36(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_12(x) + F_40(x))
Eq(F_43(x), 2*F_18(x) + F_44(x) + F_46(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_37(x) + F_43(x))
Eq(F_46(x), F_12(x)*F_39(x))
Eq(F_47(x), F_29(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_56(x))
Eq(F_51(x), F_52(x) + F_54(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_23(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_51(x))
Eq(F_56(x), F_57(x) + F_60(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_52(x) + F_57(x))
Eq(F_60(x), 3*F_18(x) + F_61(x) + F_63(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_54(x) + F_60(x))
Eq(F_63(x), F_12(x)*F_56(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_75(x))
Eq(F_66(x), F_23(x) + F_67(x))
Eq(F_67(x), F_18(x) + F_68(x) + F_74(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_72(x))
Eq(F_70(x), F_10(x) + F_71(x))
Eq(F_71(x), F_38(x))
Eq(F_72(x), F_67(x) + F_73(x))
Eq(F_73(x), F_55(x))
Eq(F_74(x), F_12(x)*F_66(x))
Eq(F_75(x), F_76(x) + F_83(x))
Eq(F_76(x), F_18(x) + F_77(x) + F_81(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_12(x))
Eq(F_80(x), F_52(x))
Eq(F_81(x), F_12(x)*F_82(x))
Eq(F_82(x), F_23(x) + F_76(x))
Eq(F_83(x), F_18(x) + F_84(x) + F_88(x) + F_90(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_37(x))
Eq(F_87(x), F_54(x))
Eq(F_88(x), F_12(x)*F_89(x))
Eq(F_89(x), F_67(x) + F_83(x))
Eq(F_90(x), F_12(x)*F_75(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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