0132_0213_1230_2301_3120
Counting sequence:
1, 1, 2, 6, 19, 50, 110, 215, 390, 677, 1151, 1952, 3349, 5868, 10548, 19453, 36696, 70487, 137229, 269710, 533487, 1059654, 2110378, 4209971, 8407034, 16798745, 33579435, 67137740, 134250905, 268473392, 536914096, 1073790777, 2147538924, 4295029483, 8590004313, 17179947098, 34359825171, 68719573162, 137439060294, 274878024975, 549755943982, 1099511770829, 2199023412503, 4398046682936, 8796093209949, 17592186249140, 35184372311660, 70368744419765, 140737488617920, 281474976995007, 562949953728741, 1125899907174502, 2251799814042999, 4503599627755598, 9007199255154978, 18014398509926443, 36028797019440546, 72057594038438337, 144115188076401859, 288230376152295140, 576460752304046177, 1152921504607510904, 2305843009214401128, 4611686018428140401, 9223372036855575764, 18446744073710401235, 36893488147420004785, 73786976294839162290, 147573952589677425435, 295147905179353897522, 590295810358706785086, 1180591620717412501127, 2361183241434823871574, 4722366482869646548213, 9444732965739291834543, 18889465931478582337488, 37778931862957163270821, 75557863725914325062012, 151115727451828648565924, 302231454903657295492205, 604462909807314589260072, 1208925819614629176707879, 2417851639229258351512253, 4835703278458516701026366, 9671406556917033399956479, 19342813113834066797715030, 38685626227668133593126810, 77371252455336267183841315, 154742504910672534365157450, 309485009821345068727672937, 618970019642690137452583131, 1237940039285380274902278652, 2475880078570760549801540649, 4951760157141521099599931328, 9903520314283042199196575008, 19807040628566084398389720233, 39614081257132168796775863996, 79228162514264337593548000187, 158456325028528675187092116489, 316912650057057350374180188170, 633825300114114700748356165667
Generating function in Maple syntax:
-(2*x^8-5*x^7-x^6+9*x^5-12*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{2 x^{8}-5 x^{7}-x^{6}+9 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-2*x**8 + 5*x**7 + x**6 - 9*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)+2*x^8-5*x^7-x^6+9*x^5-12*x^4+18*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{8}-5 x^{7}-x^{6}+9 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,4+2^(-1+n)+1/24*n^4+5/12*n^3-49/24*n^2+7/12*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 4+2^{-1+n}+\frac{n^{4}}{24}+\frac{5 n^{3}}{12}-\frac{49 n^{2}}{24}+\frac{7 n}{12} & \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) = 50
a(6) = 110
a(7) = 215
a(8) = 390
a(n+1) = 2*a(n)-1/24*(-2+n)*(n^3+8*n^2-69*n-60), n >= 9
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) = 50
a \! \left(6\right) = 110
a \! \left(7\right) = 215
a \! \left(8\right) = 390
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(-2+n \right) \left(n^{3}+8 n^{2}-69 n -60\right)}{24}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18495/
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[26,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[19,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[1,x]+F[22,x]
F[25,x] = F[20,x]
F[26,x] = F[27,x]+F[36,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[24,x]+F[30,x]
F[30,x] = F[27,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[22,x]
F[35,x] = F[31,x]
F[36,x] = F[37,x]+F[38,x]+F[67,x]
F[37,x] = 0
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]+F[61,x]
F[40,x] = F[41,x]+F[6,x]
F[41,x] = F[37,x]+F[42,x]+F[44,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[10,x]+F[25,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[21,x]+F[46,x]
F[46,x] = F[47,x]+F[54,x]
F[47,x] = F[37,x]+F[48,x]+F[50,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[12,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[22,x]
F[53,x] = F[47,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[56,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[22,x]+F[57,x]
F[60,x] = F[55,x]
F[61,x] = F[36,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[41,x]
F[66,x] = F[62,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[69,x]+F[74,x]
F[69,x] = F[22,x]+F[70,x]
F[70,x] = F[37,x]+F[71,x]+F[73,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[10,x]+F[70,x]
F[73,x] = F[12,x]*F[69,x]
F[74,x] = F[75,x]+F[86,x]
F[75,x] = F[37,x]+F[76,x]+F[79,x]
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[22,x]+F[78,x]
F[78,x] = F[76,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[81,x]+F[85,x]
F[81,x] = F[22,x]+F[82,x]
F[82,x] = F[83,x]
F[83,x] = F[12,x]*F[84,x]
F[84,x] = F[12,x]+F[82,x]
F[85,x] = F[75,x]
F[86,x] = 2*F[37,x]+F[87,x]+F[90,x]
F[87,x] = F[12,x]*F[88,x]
F[88,x] = F[25,x]+F[89,x]
F[89,x] = F[87,x]
F[90,x] = F[12,x]*F[91,x]
F[91,x] = F[57,x]+F[92,x]
F[92,x] = F[90,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_{26}\! \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_{19}\! \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_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{20}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{36}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{24}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \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_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{22}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{67}\! \left(x \right)
F_{37}\! \left(x \right) = 0
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{61}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{6}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{21}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{54}\! \left(x \right)
F_{47}\! \left(x \right) = F_{37}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{22}\! \left(x \right)
F_{53}\! \left(x \right) = F_{47}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \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_{22}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{55}\! \left(x \right)
F_{61}\! \left(x \right) = F_{36}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{41}\! \left(x \right)
F_{66}\! \left(x \right) = F_{62}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{74}\! \left(x \right)
F_{69}\! \left(x \right) = F_{22}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{37}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{10}\! \left(x \right)+F_{70}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{86}\! \left(x \right)
F_{75}\! \left(x \right) = F_{37}\! \left(x \right)+F_{76}\! \left(x \right)+F_{79}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{22}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{76}\! \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_{85}\! \left(x \right)
F_{81}\! \left(x \right) = F_{22}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right)+F_{82}\! \left(x \right)
F_{85}\! \left(x \right) = F_{75}\! \left(x \right)
F_{86}\! \left(x \right) = 2 F_{37}\! \left(x \right)+F_{87}\! \left(x \right)+F_{90}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{25}\! \left(x \right)+F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{87}\! \left(x \right)
F_{90}\! \left(x \right) = F_{12}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{57}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{90}\! \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_26(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_19(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_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_1(x) + F_22(x))
Eq(F_25(x), F_20(x))
Eq(F_26(x), F_27(x) + F_36(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_24(x) + F_30(x))
Eq(F_30(x), F_27(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_22(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_37(x) + F_38(x) + F_67(x))
Eq(F_37(x), 0)
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_61(x))
Eq(F_40(x), F_41(x) + F_6(x))
Eq(F_41(x), F_37(x) + F_42(x) + F_44(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_10(x) + F_25(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_21(x) + F_46(x))
Eq(F_46(x), F_47(x) + F_54(x))
Eq(F_47(x), F_37(x) + F_48(x) + F_50(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_12(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_22(x))
Eq(F_53(x), F_47(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_56(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_22(x) + F_57(x))
Eq(F_60(x), F_55(x))
Eq(F_61(x), F_36(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_41(x))
Eq(F_66(x), F_62(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_74(x))
Eq(F_69(x), F_22(x) + F_70(x))
Eq(F_70(x), F_37(x) + F_71(x) + F_73(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_10(x) + F_70(x))
Eq(F_73(x), F_12(x)*F_69(x))
Eq(F_74(x), F_75(x) + F_86(x))
Eq(F_75(x), F_37(x) + F_76(x) + F_79(x))
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_22(x) + F_78(x))
Eq(F_78(x), F_76(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_81(x) + F_85(x))
Eq(F_81(x), F_22(x) + F_82(x))
Eq(F_82(x), F_83(x))
Eq(F_83(x), F_12(x)*F_84(x))
Eq(F_84(x), F_12(x) + F_82(x))
Eq(F_85(x), F_75(x))
Eq(F_86(x), 2*F_37(x) + F_87(x) + F_90(x))
Eq(F_87(x), F_12(x)*F_88(x))
Eq(F_88(x), F_25(x) + F_89(x))
Eq(F_89(x), F_87(x))
Eq(F_90(x), F_12(x)*F_91(x))
Eq(F_91(x), F_57(x) + F_92(x))
Eq(F_92(x), F_90(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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