0123_0132_1203_1230_2301_3012
Counting sequence:
1, 1, 2, 6, 18, 48, 115, 255, 537, 1095, 2192, 4348, 8596, 16994, 33649, 66765, 132739, 264353, 527158, 1052242, 2101766, 4200036, 8395647, 16785771, 33564733, 67121163, 134232300, 268452600, 536890952, 1073765110, 2147510557, 4294998233, 8589969991, 17179909509, 34359784114, 68719528430, 137439011674, 274877972248, 549755886923, 1099511709207, 2199023346081, 4398046611471, 8796093133192, 17592186166836, 35184372223548, 70368744325578, 140737488517385, 281474976887845, 562949953614667, 1125899907053225, 2251799813914222, 4503599627619018, 9007199255010286, 18014398509773324, 36028797019278679, 72057594038267395, 144115188076221509, 288230376152105043, 576460752303845988, 1152921504607300272, 2305843009214179696, 4611686018427907806, 9223372036855331637, 18446744073710145201, 36893488147419736463, 73786976294838881293, 147573952589677131370, 295147905179353589990, 590295810358706463682, 1180591620717412165440, 2361183241434823521187, 4722366482869646182703, 9444732965739291453481, 18889465931478581940439, 37778931862957162857344, 75557863725914324631660, 151115727451828648118244, 302231454903657295026738, 604462909807314588776353, 1208925819614629176205437, 2417851639229258350990611, 4835703278458516700485041, 9671406556917033399394982, 19342813113834066797132866, 38685626227668133592523478, 77371252455336267183216308, 154742504910672534364510255, 309485009821345068727003035, 618970019642690137451889997, 1237940039285380274901561755, 2475880078570760549800799452, 4951760157141521099599165288, 9903520314283042199195783576, 19807040628566084398388902854, 39614081257132168796775020109, 79228162514264337593547129225, 158456325028528675187091217879, 316912650057057350374179261333, 633825300114114700748355210018, 1267650600228229401496706964702, 2535301200456458802993410326826
Generating function in Maple syntax:
-(4*x^5-11*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{4 x^{5}-11 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:
(-4*x**5 + 11*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)+4*x^5-11*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)+4 x^{5}-11 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
-1+2^(n+1)-1/24*n^2-7/4*n-1/4*n^3+1/24*n^4
Explicit closed form in latex syntax:
-1+2^{n +1}-\frac{n^{2}}{24}-\frac{7 n}{4}-\frac{n^{3}}{4}+\frac{n^{4}}{24}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 48
a(n+1) = 2*a(n)-1/24*(n-1)*(n^3-9*n^2+2*n-24), n >= 6
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) = 18
a \! \left(5\right) = 48
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{3}-9 n^{2}+2 n -24\right)}{24}, \quad n \geq 6
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://www.permpal.com/tree/22096/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[11,x]*F[3,x]
F[3,x] = F[1,x]+F[33,x]+F[4,x]
F[4,x] = F[11,x]*F[5,x]
F[5,x] = F[19,x]+F[6,x]
F[6,x] = F[1,x]+F[4,x]+F[7,x]
F[7,x] = F[11,x]*F[8,x]
F[8,x] = F[12,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]*F[9,x]
F[11,x] = x
F[12,x] = F[13,x]+F[14,x]
F[13,x] = F[10,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[11,x]*F[17,x]
F[17,x] = F[13,x]+F[14,x]
F[18,x] = F[11,x]*F[12,x]
F[19,x] = F[20,x]
F[20,x] = F[11,x]*F[21,x]
F[21,x] = F[20,x]+F[22,x]+F[23,x]
F[22,x] = F[9,x]^2
F[23,x] = F[24,x]
F[24,x] = F[11,x]*F[25,x]*F[9,x]
F[25,x] = F[12,x]+F[26,x]
F[26,x] = F[27,x]*F[9,x]
F[27,x] = F[28,x]+F[9,x]
F[28,x] = F[13,x]+F[29,x]
F[29,x] = F[15,x]+F[30,x]+F[32,x]
F[30,x] = F[11,x]*F[31,x]
F[31,x] = F[13,x]+F[29,x]
F[32,x] = F[11,x]*F[28,x]
F[33,x] = F[11,x]*F[34,x]
F[34,x] = F[35,x]+F[54,x]
F[35,x] = F[36,x]+F[48,x]
F[36,x] = F[1,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[11,x]*F[39,x]
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[1,x]+F[11,x]
F[41,x] = F[37,x]+F[42,x]
F[42,x] = F[15,x]+F[43,x]+F[47,x]
F[43,x] = F[11,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[11,x]
F[46,x] = F[42,x]
F[47,x] = F[11,x]*F[13,x]
F[48,x] = F[13,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[11,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[13,x]
F[53,x] = F[49,x]
F[54,x] = F[55,x]+F[76,x]
F[55,x] = F[13,x]+F[56,x]
F[56,x] = F[15,x]+F[57,x]+F[59,x]
F[57,x] = F[11,x]*F[58,x]
F[58,x] = F[37,x]+F[56,x]
F[59,x] = F[11,x]*F[60,x]
F[60,x] = F[61,x]+F[65,x]
F[61,x] = F[13,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[11,x]*F[64,x]
F[64,x] = F[11,x]+F[62,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[15,x]+F[16,x]+F[59,x]
F[67,x] = 2*F[15,x]+F[68,x]+F[72,x]
F[68,x] = F[11,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[47,x]
F[71,x] = F[68,x]
F[72,x] = F[11,x]*F[73,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[62,x]
F[75,x] = F[67,x]
F[76,x] = F[77,x]+F[83,x]
F[77,x] = F[15,x]+F[78,x]+F[80,x]
F[78,x] = F[11,x]*F[79,x]
F[79,x] = F[13,x]+F[77,x]
F[80,x] = F[11,x]*F[81,x]
F[81,x] = F[13,x]+F[82,x]
F[82,x] = F[80,x]
F[83,x] = 2*F[15,x]+F[84,x]+F[86,x]
F[84,x] = F[11,x]*F[85,x]
F[85,x] = F[49,x]+F[83,x]
F[86,x] = F[11,x]*F[87,x]
F[87,x] = F[88,x]+F[89,x]
F[88,x] = F[82,x]
F[89,x] = F[86,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_{11}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{33}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{11}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{19}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{11}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{12}\! \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_{9}\! \left(x \right)
F_{11}\! \left(x \right) = x
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{11}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right) F_{12}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{20}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{9} \left(x \right)^{2}
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right) F_{25}\! \left(x \right) F_{9}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{9}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{9}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{11}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right) F_{28}\! \left(x \right)
F_{33}\! \left(x \right) = F_{11}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{54}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{48}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{11}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{43}\! \left(x \right) = F_{11}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{11}\! \left(x \right)
F_{46}\! \left(x \right) = F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right)
F_{48}\! \left(x \right) = F_{13}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{11}\! \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_{13}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{76}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{15}\! \left(x \right)+F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{11}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{37}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{11}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{65}\! \left(x \right)
F_{61}\! \left(x \right) = F_{13}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{11}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{11}\! \left(x \right)+F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{59}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{68}\! \left(x \right)+F_{72}\! \left(x \right)
F_{68}\! \left(x \right) = F_{11}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{47}\! \left(x \right)
F_{71}\! \left(x \right) = F_{68}\! \left(x \right)
F_{72}\! \left(x \right) = F_{11}\! \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_{62}\! \left(x \right)
F_{75}\! \left(x \right) = F_{67}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{83}\! \left(x \right)
F_{77}\! \left(x \right) = F_{15}\! \left(x \right)+F_{78}\! \left(x \right)+F_{80}\! \left(x \right)
F_{78}\! \left(x \right) = F_{11}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{13}\! \left(x \right)+F_{77}\! \left(x \right)
F_{80}\! \left(x \right) = F_{11}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{13}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{80}\! \left(x \right)
F_{83}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{84}\! \left(x \right)+F_{86}\! \left(x \right)
F_{84}\! \left(x \right) = F_{11}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{49}\! \left(x \right)+F_{83}\! \left(x \right)
F_{86}\! \left(x \right) = F_{11}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)+F_{89}\! \left(x \right)
F_{88}\! \left(x \right) = F_{82}\! \left(x \right)
F_{89}\! \left(x \right) = F_{86}\! \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_11(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_33(x) + F_4(x))
Eq(F_4(x), F_11(x)*F_5(x))
Eq(F_5(x), F_19(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_4(x) + F_7(x))
Eq(F_7(x), F_11(x)*F_8(x))
Eq(F_8(x), F_12(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x)*F_9(x))
Eq(F_11(x), x)
Eq(F_12(x), F_13(x) + F_14(x))
Eq(F_13(x), F_10(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_11(x)*F_17(x))
Eq(F_17(x), F_13(x) + F_14(x))
Eq(F_18(x), F_11(x)*F_12(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_11(x)*F_21(x))
Eq(F_21(x), F_20(x) + F_22(x) + F_23(x))
Eq(F_22(x), F_9(x)**2)
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_11(x)*F_25(x)*F_9(x))
Eq(F_25(x), F_12(x) + F_26(x))
Eq(F_26(x), F_27(x)*F_9(x))
Eq(F_27(x), F_28(x) + F_9(x))
Eq(F_28(x), F_13(x) + F_29(x))
Eq(F_29(x), F_15(x) + F_30(x) + F_32(x))
Eq(F_30(x), F_11(x)*F_31(x))
Eq(F_31(x), F_13(x) + F_29(x))
Eq(F_32(x), F_11(x)*F_28(x))
Eq(F_33(x), F_11(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_54(x))
Eq(F_35(x), F_36(x) + F_48(x))
Eq(F_36(x), F_1(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_11(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_1(x) + F_11(x))
Eq(F_41(x), F_37(x) + F_42(x))
Eq(F_42(x), F_15(x) + F_43(x) + F_47(x))
Eq(F_43(x), F_11(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_11(x))
Eq(F_46(x), F_42(x))
Eq(F_47(x), F_11(x)*F_13(x))
Eq(F_48(x), F_13(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_11(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_13(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), F_55(x) + F_76(x))
Eq(F_55(x), F_13(x) + F_56(x))
Eq(F_56(x), F_15(x) + F_57(x) + F_59(x))
Eq(F_57(x), F_11(x)*F_58(x))
Eq(F_58(x), F_37(x) + F_56(x))
Eq(F_59(x), F_11(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_65(x))
Eq(F_61(x), F_13(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_11(x)*F_64(x))
Eq(F_64(x), F_11(x) + F_62(x))
Eq(F_65(x), F_66(x) + F_67(x))
Eq(F_66(x), F_15(x) + F_16(x) + F_59(x))
Eq(F_67(x), 2*F_15(x) + F_68(x) + F_72(x))
Eq(F_68(x), F_11(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_47(x))
Eq(F_71(x), F_68(x))
Eq(F_72(x), F_11(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_62(x))
Eq(F_75(x), F_67(x))
Eq(F_76(x), F_77(x) + F_83(x))
Eq(F_77(x), F_15(x) + F_78(x) + F_80(x))
Eq(F_78(x), F_11(x)*F_79(x))
Eq(F_79(x), F_13(x) + F_77(x))
Eq(F_80(x), F_11(x)*F_81(x))
Eq(F_81(x), F_13(x) + F_82(x))
Eq(F_82(x), F_80(x))
Eq(F_83(x), 2*F_15(x) + F_84(x) + F_86(x))
Eq(F_84(x), F_11(x)*F_85(x))
Eq(F_85(x), F_49(x) + F_83(x))
Eq(F_86(x), F_11(x)*F_87(x))
Eq(F_87(x), F_88(x) + F_89(x))
Eq(F_88(x), F_82(x))
Eq(F_89(x), F_86(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_and_col_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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