0132_0231_1320
Counting sequence:
1, 1, 2, 6, 21, 76, 274, 979, 3479, 12351, 43951, 157081, 564409, 2039465, 7410650, 27070098, 99369477, 366413428, 1356673303, 5042049056, 18802857995, 70339266486, 263885350191, 992602334038, 3742727924011, 14144071435144, 53562729072965, 203230711375158, 772497640716787, 2941268143229968, 11216441560211794, 42836730705967091, 163824862676759807, 627350838044214903, 2405338568811541003, 9233100318576674509, 35481127702924224505, 136490000023634870917, 525575068618446963875, 2025712652523404915211, 7814663816151834168333, 30172670243050733491207, 116592621913135835371345, 450885545294475883979591, 1744956401584931935048433, 6757902765947011142644347, 26189990315114769100247535, 101564640969443087674604281, 394114163348383219784704705, 1530252302670999950300374241, 5945047874778021044362651625, 23109438354382637943792800109, 89878556463993634499135147781, 349741279071526425801039034525, 1361613078571360472904628436113, 5303577110173462811724010461377, 20667367064842235160718568152301, 80574129521843340294327738346281, 314263364809379002713227020281189, 1226233960177670518612275631734025, 4786614605725188225899640931676101, 18691902012352125359036788269935681, 73020210624368727601514682850557482, 285359350665022928130486831730711842, 1115567105905462753847041883125843173, 4362638235880906277153698048924388596, 17066645353584688047007885320254493103, 66786446158060141321060314383243188328, 261436152541740265289628296457519826139, 1023708231811935590108770044758277047022, 4009740417998037589252228930669234398699, 15710219545216069583947396524061618609114, 61570226642321596508140035006307134472555, 241367315124426510632670064573643078179988, 946459891198832406692254288006691912036525, 3712265069278715741997106695401406926198850, 14564173146369708353995554456791481562055931, 57153123445653417433997449928887886078862484, 224335852266034304314771261840069791398385451, 880763646342554140143245752503937010663514196, 3458753000091180341002612216293503687924956007, 13585535953333630745001689473584521506499156610, 53373875142470906800071912178496004480194118517, 209736097019513047264323445770677850096444033052, 824342992098984253739343567939557938086040304383, 3240641205086344040895867262609864890563693868386, 12742075140836377929496539064847651153225198799113, 50111065379022620000930462269829455155618131151348, 197110376667917195876035271306250564950349419736739, 775471595602452907220168766258622231563409925585602, 3051413720631931347765818095216232934777692232904609, 12009180769802349484974430802301744291081056448278104, 47271687587589529208514185679231884087254176328529975, 186106983784908135067967004852604751339080180559955798, 732818671011445180527666050401215835943682547687413975, 2886032205265883003500165345931164644481899620760549310, 11367766876364317730355748455538184397550159428916081411, 44783401161893247085040010217740620921496637304493066944, 176451580101236179066031344012246278170297130166496581781, 695343287956826632095297228359458249945343616449804796246, 2740544967809346542756770034823036493456951938082380030747
Generating function in Maple syntax:
1/4*((-6*x^3+9*x^2-5*x+1)*(1-4*x)^(1/2)-10*x^4+24*x^3-19*x^2+7*x-1)/(x-1/2)/x^2/(-1+x)^2
Generating function in latex syntax:
\frac{\left(-6 x^{3}+9 x^{2}-5 x +1\right) \sqrt{1-4 x}-10 x^{4}+24 x^{3}-19 x^{2}+7 x -1}{4 \left(x -\frac{1}{2}\right) x^{2} \left(-1+x \right)^{2}}
Generating function in sympy syntax:
(-5*x**4/2 + 6*x**3 - 19*x**2/4 + 7*x/4 + sqrt(1 - 4*x)*(-6*x**3 + 9*x**2 - 5*x + 1)/4 - 1/4)/(x**2*(x - 1)**2*(x - 1/2))
Implicit equation for the generating function in Maple syntax:
x^2*(2*x-1)^2*(-1+x)^4*F(x)^2+(2*x-1)*(10*x^4-24*x^3+19*x^2-7*x+1)*(-1+x)^2*F(x)+25*x^6-84*x^5+122*x^4-95*x^3+42*x^2-10*x+1 = 0
Implicit equation for the generating function in latex syntax:
x^{2} \left(2 x -1\right)^{2} \left(-1+x \right)^{4} F \! \left(x \right)^{2}+\left(2 x -1\right) \left(10 x^{4}-24 x^{3}+19 x^{2}-7 x +1\right) \left(-1+x \right)^{2} F \! \left(x \right)+25 x^{6}-84 x^{5}+122 x^{4}-95 x^{3}+42 x^{2}-10 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 76
a(6) = 274
a(n+6) = -24*(2*n+3)/(8+n)*a(n)-2*(104*n+335)/(8+n)*a(2+n)+12*(13*n+29)/(8+n)*a(n+1)+3*(49*n+213)/(8+n)*a(n+3)-(58*n+321)/(8+n)*a(n+4)+3*(4*n+27)/(8+n)*a(n+5)+7/(8+n), 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) = 21
a \! \left(5\right) = 76
a \! \left(6\right) = 274
a \! \left(n +6\right) = -\frac{24 \left(2 n +3\right) a \! \left(n \right)}{8+n}-\frac{2 \left(104 n +335\right) a \! \left(2+n \right)}{8+n}+\frac{12 \left(13 n +29\right) a \! \left(n +1\right)}{8+n}+\frac{3 \left(49 n +213\right) a \! \left(n +3\right)}{8+n}-\frac{\left(58 n +321\right) a \! \left(n +4\right)}{8+n}+\frac{3 \left(4 n +27\right) a \! \left(n +5\right)}{8+n}+\frac{7}{8+n}, \quad n \geq 7
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/268/
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[4,x]+F[72,x]
F[4,x] = F[11,x]*F[5,x]
F[5,x] = F[1,x]+F[12,x]+F[4,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[11,x]*F[5,x]*F[8,x]
F[8,x] = F[1,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[8,x]^2*F[11,x]
F[11,x] = x
F[12,x] = F[11,x]*F[13,x]
F[13,x] = F[1,x]+F[12,x]+F[14,x]+F[16,x]+F[57,x]
F[14,x] = F[11,x]*F[15,x]
F[15,x] = F[1,x]+F[12,x]+F[14,x]+F[4,x]
F[16,x] = F[11,x]*F[17,x]
F[17,x] = F[1,x]+F[12,x]+F[16,x]+F[18,x]+F[20,x]
F[18,x] = F[19,x]
F[19,x] = F[11,x]*F[17,x]*F[8,x]
F[20,x] = F[21,x]
F[21,x] = F[11,x]*F[22,x]
F[22,x] = F[23,x]+F[54,x]
F[23,x] = F[24,x]+F[40,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[11,x]*F[27,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[11,x]*F[28,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[33,x]
F[33,x] = F[11,x]*F[34,x]
F[34,x] = F[1,x]+F[32,x]
F[35,x] = F[36,x]+F[37,x]+F[39,x]
F[36,x] = 0
F[37,x] = F[11,x]*F[38,x]
F[38,x] = F[29,x]+F[35,x]
F[39,x] = F[11,x]*F[31,x]
F[40,x] = F[32,x]+F[41,x]
F[41,x] = F[36,x]+F[42,x]+F[44,x]
F[42,x] = F[11,x]*F[43,x]
F[43,x] = F[25,x]+F[41,x]
F[44,x] = F[11,x]*F[45,x]
F[45,x] = F[31,x]+F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[48,x]
F[48,x] = F[11,x]*F[49,x]
F[49,x] = F[32,x]+F[47,x]
F[50,x] = 2*F[36,x]+F[51,x]+F[53,x]
F[51,x] = F[11,x]*F[52,x]
F[52,x] = F[35,x]+F[50,x]
F[53,x] = F[11,x]*F[46,x]
F[54,x] = F[55,x]
F[55,x] = F[11,x]*F[34,x]*F[56,x]
F[56,x] = F[24,x]+F[25,x]
F[57,x] = F[58,x]
F[58,x] = F[11,x]*F[59,x]
F[59,x] = F[60,x]+F[70,x]
F[60,x] = F[23,x]+F[61,x]
F[61,x] = F[62,x]+F[66,x]
F[62,x] = F[32,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[11,x]*F[65,x]
F[65,x] = F[25,x]+F[63,x]
F[66,x] = F[47,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[11,x]*F[69,x]
F[69,x] = F[63,x]+F[67,x]
F[70,x] = F[71,x]
F[71,x] = F[34,x]^2*F[11,x]*F[27,x]
F[72,x] = F[11,x]*F[73,x]
F[73,x] = F[1,x]+F[16,x]+F[20,x]+F[72,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_{4}\! \left(x \right)+F_{72}\! \left(x \right)
F_{4}\! \left(x \right) = F_{11}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{4}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{11}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{8} \left(x \right)^{2} F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = x
F_{12}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{14}\! \left(x \right)+F_{16}\! \left(x \right)+F_{57}\! \left(x \right)
F_{14}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{14}\! \left(x \right)+F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)+F_{20}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \left(x \right) F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{54}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{40}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{11}\! \left(x \right) F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{11}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = 0
F_{37}\! \left(x \right) = F_{11}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{29}\! \left(x \right)+F_{35}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right) F_{31}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{36}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{25}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{31}\! \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_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{11}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{32}\! \left(x \right)+F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{36}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{11}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{35}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \left(x \right) F_{46}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{11}\! \left(x \right) F_{34}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{11}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{70}\! \left(x \right)
F_{60}\! \left(x \right) = F_{23}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{66}\! \left(x \right)
F_{62}\! \left(x \right) = F_{32}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{11}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{25}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{47}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{11}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{63}\! \left(x \right)+F_{67}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{34} \left(x \right)^{2} F_{11}\! \left(x \right) F_{27}\! \left(x \right)
F_{72}\! \left(x \right) = F_{11}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)+F_{72}\! \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_4(x) + F_72(x))
Eq(F_4(x), F_11(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_12(x) + F_4(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_11(x)*F_5(x)*F_8(x))
Eq(F_8(x), F_1(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_8(x)**2)
Eq(F_11(x), x)
Eq(F_12(x), F_11(x)*F_13(x))
Eq(F_13(x), F_1(x) + F_12(x) + F_14(x) + F_16(x) + F_57(x))
Eq(F_14(x), F_11(x)*F_15(x))
Eq(F_15(x), F_1(x) + F_12(x) + F_14(x) + F_4(x))
Eq(F_16(x), F_11(x)*F_17(x))
Eq(F_17(x), F_1(x) + F_12(x) + F_16(x) + F_18(x) + F_20(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_11(x)*F_17(x)*F_8(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_11(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_54(x))
Eq(F_23(x), F_24(x) + F_40(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_11(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_11(x)*F_28(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_11(x)*F_34(x))
Eq(F_34(x), F_1(x) + F_32(x))
Eq(F_35(x), F_36(x) + F_37(x) + F_39(x))
Eq(F_36(x), 0)
Eq(F_37(x), F_11(x)*F_38(x))
Eq(F_38(x), F_29(x) + F_35(x))
Eq(F_39(x), F_11(x)*F_31(x))
Eq(F_40(x), F_32(x) + F_41(x))
Eq(F_41(x), F_36(x) + F_42(x) + F_44(x))
Eq(F_42(x), F_11(x)*F_43(x))
Eq(F_43(x), F_25(x) + F_41(x))
Eq(F_44(x), F_11(x)*F_45(x))
Eq(F_45(x), F_31(x) + F_46(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_11(x)*F_49(x))
Eq(F_49(x), F_32(x) + F_47(x))
Eq(F_50(x), 2*F_36(x) + F_51(x) + F_53(x))
Eq(F_51(x), F_11(x)*F_52(x))
Eq(F_52(x), F_35(x) + F_50(x))
Eq(F_53(x), F_11(x)*F_46(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_11(x)*F_34(x)*F_56(x))
Eq(F_56(x), F_24(x) + F_25(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_11(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_70(x))
Eq(F_60(x), F_23(x) + F_61(x))
Eq(F_61(x), F_62(x) + F_66(x))
Eq(F_62(x), F_32(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_11(x)*F_65(x))
Eq(F_65(x), F_25(x) + F_63(x))
Eq(F_66(x), F_47(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_11(x)*F_69(x))
Eq(F_69(x), F_63(x) + F_67(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_11(x)*F_27(x)*F_34(x)**2)
Eq(F_72(x), F_11(x)*F_73(x))
Eq(F_73(x), F_1(x) + F_16(x) + F_20(x) + F_72(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "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_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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