0132_1302_2130_3021
Counting sequence:
1, 1, 2, 6, 20, 63, 184, 509, 1364, 3597, 9421, 24621, 64332, 168166, 439815, 1150733, 3011579, 7882864, 20635449, 54021392, 141425991, 370253066, 969328762, 2537727676, 6643847435, 17393806303, 45537561424, 119218865944, 312119022134, 817138183637, 2139295509086, 5600748320711, 14662949426542, 38388099928411, 100501350323755, 263115951003023, 688846502640094, 1803423556866124, 4721424167900669, 12360848946771207, 32361122672340581, 84722519070169806, 221806434538079047, 580696784543967746, 1520283919093714025, 3980154972737052768, 10420180999117310464, 27280388024614731654, 71420983074726723429, 186982561199565262477, 489526700523968871726, 1281597540372341143226, 3355265920593054330152, 8784200221406821599931, 22997334743627410201620, 60207804009475408714913, 157626077284798815629784, 412670427844921037836409, 1080385206249964297515289, 2828485190904971854317697, 7405070366464951265016896, 19386725908489881940281346, 50755107359004694555343107, 132878596168524201725229841, 347880681146567910619792415, 910763447271179530133555708, 2384409660666970679780243429, 6242465534729732509206501764, 16342986943522226847838545499, 42786495295836948034308372742, 112016498943988617255085762966, 293263001536128903730948056416, 767772505664398093937757494287, 2010054515457065378082323459851, 5262391040706798040309211861660, 13777118606663328742845311042028, 36068964779283188188226720119274, 94429775731186235821834848105969, 247220362414275519277277822921434, 647231311511640322009998619310987, 1694473572120645446752718033591186, 4436189404850296018248155479966311, 11614094642430242607991748404732567, 30406094522440431805727089732574211, 79604188924891052809189520791247730, 208406472252232726621841472639338248, 545615227831807127056334897124844569, 1428439211243188654547163218733177899, 3739702405897758836585154759072572969, 9790668006450087855208301058482322682, 25632301613452504729039748416372070931, 67106236833907426331910944190631456406, 175686408888269774266693084155519751197, 459952989830901896468168308275925132796, 1204172560604435915137811840672252861500, 3152564691982405848945267213740830540618, 8253521515342781631697989800550235719689, 21607999854045939046148702187909873443929, 56570478046795035506748116763179381299354, 148103434286339167474095648101628266998702, 387739824812222466915538827541705416094076
Generating function in Maple syntax:
-(2*x^6-12*x^5+24*x^4-29*x^3+20*x^2-7*x+1)/(x^2-3*x+1)/(x-1)^5
Generating function in latex syntax:
-\frac{2 x^{6}-12 x^{5}+24 x^{4}-29 x^{3}+20 x^{2}-7 x +1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-2*x**6 + 12*x**5 - 24*x**4 + 29*x**3 - 20*x**2 + 7*x - 1)/((x - 1)**5*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^5*F(x)+2*x^6-12*x^5+24*x^4-29*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{6}-12 x^{5}+24 x^{4}-29 x^{3}+20 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
1/24*(12*5^(1/2)-12)*(3/2-1/2*5^(1/2))^(-n)+1/24*(-12*5^(1/2)-12)*(3/2+1/2*5^(1/2))^(-n)+1/24*n^4-5/12*n^3+23/24*n^2-31/12*n+2
Explicit closed form in latex syntax:
\frac{\left(12 \sqrt{5}-12\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{24}+\frac{\left(-12 \sqrt{5}-12\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{24}+\frac{n^{4}}{24}-\frac{5 n^{3}}{12}+\frac{23 n^{2}}{24}-\frac{31 n}{12}+2
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(n+2) = -a(n)+3*a(n+1)-1/24*n*(n+1)*(n^2-7*n-6), 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) = 20
a \! \left(5\right) = 63
a \! \left(6\right) = 184
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-\frac{n \left(n +1\right) \left(n^{2}-7 n -6\right)}{24}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15663/
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[31,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[20,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[17,x]
F[20,x] = F[21,x]+F[22,x]+F[30,x]
F[21,x] = 0
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[24,x]
F[27,x] = F[20,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[12,x]*F[13,x]
F[31,x] = F[32,x]+F[43,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[1,x]+F[14,x]
F[36,x] = F[32,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[32,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[40,x]
F[43,x] = F[21,x]+F[44,x]+F[69,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[54,x]
F[46,x] = F[47,x]+F[6,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[24,x]+F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[48,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[50,x]
F[54,x] = F[43,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[59,x]+F[67,x]
F[59,x] = F[21,x]+F[60,x]+F[65,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[10,x]+F[51,x]
F[63,x] = F[59,x]+F[64,x]
F[64,x] = F[56,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[32,x]+F[59,x]
F[67,x] = F[68,x]
F[68,x] = F[12,x]*F[58,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[66,x]+F[71,x]
F[71,x] = F[72,x]+F[79,x]
F[72,x] = F[73,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[14,x]+F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[24,x]
F[79,x] = 2*F[21,x]+F[80,x]+F[89,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[82,x]
F[82,x] = F[51,x]+F[83,x]
F[83,x] = F[84,x]
F[84,x] = F[12,x]*F[85,x]
F[85,x] = F[86,x]
F[86,x] = F[25,x]+F[87,x]
F[87,x] = F[88,x]
F[88,x] = F[12,x]*F[86,x]
F[89,x] = F[12,x]*F[71,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_{31}\! \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_{20}\! \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_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{43}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \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_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \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) = F_{21}\! \left(x \right)+F_{44}\! \left(x \right)+F_{69}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{54}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{6}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{24}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{48}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{54}\! \left(x \right) = F_{43}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{67}\! \left(x \right)
F_{59}\! \left(x \right) = F_{21}\! \left(x \right)+F_{60}\! \left(x \right)+F_{65}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{10}\! \left(x \right)+F_{51}\! \left(x \right)
F_{63}\! \left(x \right) = F_{59}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{56}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{32}\! \left(x \right)+F_{59}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{66}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{79}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{14}\! \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_{24}\! \left(x \right)
F_{79}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{80}\! \left(x \right)+F_{89}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{51}\! \left(x \right)+F_{83}\! \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_{86}\! \left(x \right) = F_{25}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{12}\! \left(x \right) F_{86}\! \left(x \right)
F_{89}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \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_31(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_20(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_17(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_30(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_24(x))
Eq(F_27(x), F_20(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_27(x))
Eq(F_30(x), F_12(x)*F_13(x))
Eq(F_31(x), F_32(x) + F_43(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_1(x) + F_14(x))
Eq(F_36(x), F_32(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_32(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_40(x))
Eq(F_43(x), F_21(x) + F_44(x) + F_69(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_54(x))
Eq(F_46(x), F_47(x) + F_6(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_24(x) + F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_48(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_50(x))
Eq(F_54(x), F_43(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_59(x) + F_67(x))
Eq(F_59(x), F_21(x) + F_60(x) + F_65(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_10(x) + F_51(x))
Eq(F_63(x), F_59(x) + F_64(x))
Eq(F_64(x), F_56(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_32(x) + F_59(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_12(x)*F_58(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_66(x) + F_71(x))
Eq(F_71(x), F_72(x) + F_79(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_14(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_24(x))
Eq(F_79(x), 2*F_21(x) + F_80(x) + F_89(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_51(x) + F_83(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))
Eq(F_86(x), F_25(x) + F_87(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_12(x)*F_86(x))
Eq(F_89(x), F_12(x)*F_71(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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