0123_0213_1032_1302_1320
Counting sequence:
1, 1, 2, 6, 19, 55, 153, 416, 1115, 2962, 7825, 20601, 54121, 141994, 372237, 975326, 2554727, 6690451, 17519209, 45871356, 120101623, 314444458, 823249461, 2155332581, 5642794649, 14773126390, 38676705913, 101257187766, 265095175195, 694028852047, 1816992212985, 4756949133176, 12453857364851, 32604626485954, 85360027795897, 223475466129201, 585066385522057, 1531723714594786, 4010104797350469, 10498590740702606, 27485667527091503, 71958412006152043, 188389568759278921, 493210294705179156, 1291241316057667279, 3380513654602725850, 8850299649586822173, 23170385297128955741, 60660856246607572025, 158812183450472502382, 415775694117396204145, 1088514898922081121126, 2849769002681798439331, 7460792109176630488039, 19532607324934360596057, 51137029865766035162576, 133878482272589596325387, 350498416952368189109746, 917616768585106257733729, 2402351888803907306117481, 6289438897828163669374633, 16465964804683088432788378, 43108455516225154368528381, 112859401743998932143116606, 295469749715782252270679159, 773549847403364992349098435, 2025179792494340502666651433, 5301989530079701461221068716, 13880788797744836604456802855, 36340376863154926021179800842, 95140341791720131851573308805, 249080648512005777595061295701, 652101603744297699387622457561, 1707224162720888127083339126374, 4469570884418367986831939850217, 11701488490534217944897558402326, 30634894587184289264315358263467, 80203195271018655375988217272831, 209974691225871685808043617346489, 549720878406596416520476659442856, 1439187943993917587170114709449763, 3767842953575156382878929842050338, 9864340916731551622772465538312841, 25825179796619498584633319867643681, 67611198473126944291628137880985289, 177008415622761334549946590686434770, 463414048395157059778407774905808693, 1213233729562709845465168371669603566, 3176287140292972477717185118469103935, 8315627691316207589466366399742422427, 21770595933655650293561981275128979465, 56996160109650743295879624036020046276, 149217884395296579601617004637677505791, 390657493076238995521171550292134347834, 1022754594833420406981637920458593760877, 2677606291424022225455682645718637034701, 7010064279438646269437090725552175666297, 18352586546891916582939210674427738387166, 48047695361237103479515843150075746241249, 125790499536819393855827241771634055505606, 329323803249221078088320107013005682190643
Generating function in Maple syntax:
(x^7-2*x^5-x^4+2*x^3+4*x^2-4*x+1)/(x-1)/(x^2-3*x+1)/(x^2+x-1)
Generating function in latex syntax:
\frac{x^{7}-2 x^{5}-x^{4}+2 x^{3}+4 x^{2}-4 x +1}{\left(x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right)}
Generating function in sympy syntax:
(x**7 - 2*x**5 - x**4 + 2*x**3 + 4*x**2 - 4*x + 1)/((x - 1)*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(x^2-3*x+1)*(x^2+x-1)*F(x)+x^7-2*x^5-x^4+2*x^3+4*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+x^{7}-2 x^{5}-x^{4}+2 x^{3}+4 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,17/2*(3/2-1/2*5^(1/2))^(1-n)+17/2*(3/2+1/2*5^(1/2))^(1-n)+17/2*(-1/2*5^(1/2)-1/2)^(1-n)+17/2*(1/2*5^(1/2)-1/2)^(1-n)+1/20*(121*5^(1/2)-325)*(3/2-1/2*5^(1/2))^(-n)+1/20*(-121*5^(1/2)-325)*(3/2+1/2*5^(1/2))^(-n)+1/20*(89*5^(1/2)+75)*(-1/2*5^(1/2)-1/2)^(-n)+1+1/20*(-89*5^(1/2)+75)*(1/2*5^(1/2)-1/2)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{17 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{1-n}}{2}+\frac{17 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{1-n}}{2}+\frac{17 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{1-n}}{2}+\frac{17 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{1-n}}{2}\\+\frac{\left(121 \sqrt{5}-325\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(-121 \sqrt{5}-325\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\\\frac{\left(89 \sqrt{5}+75\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}+1+\frac{\left(-89 \sqrt{5}+75\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20} & \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) = 55
a(6) = 153
a(7) = 416
a(n+4) = a(n)-2*a(n+1)-3*a(n+2)+4*a(n+3)+1, n >= 8
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) = 55
a \! \left(6\right) = 153
a \! \left(7\right) = 416
a \! \left(n +4\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right)+1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16886/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[17,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[74,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[37,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[1,x]+F[4,x]
F[26,x] = F[19,x]+F[27,x]+F[35,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[30,x]+F[4,x]
F[30,x] = F[31,x]
F[31,x] = F[11,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[4,x]*F[7,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[33,x]+F[7,x]
F[37,x] = F[38,x]+F[65,x]
F[38,x] = F[19,x]+F[20,x]+F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[19,x]+F[42,x]+F[64,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[46,x]
F[44,x] = F[4,x]+F[45,x]
F[45,x] = F[19,x]+F[27,x]+F[34,x]
F[46,x] = F[41,x]+F[47,x]
F[47,x] = F[19,x]+F[48,x]+F[61,x]+F[63,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[57,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[4,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[33,x]
F[57,x] = F[58,x]
F[58,x] = 2*F[19,x]+F[48,x]+F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[41,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[41,x]
F[63,x] = 0
F[64,x] = F[2,x]*F[4,x]
F[65,x] = F[19,x]+F[61,x]+F[66,x]+F[73,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]+F[71,x]
F[68,x] = F[69,x]+F[70,x]
F[69,x] = F[52,x]
F[70,x] = F[55,x]
F[71,x] = F[72,x]
F[72,x] = 2*F[19,x]+F[59,x]+F[66,x]
F[73,x] = 0
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]
F[76,x] = F[2,x]+F[77,x]
F[77,x] = F[19,x]+F[42,x]+F[78,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[2,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{17}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{74}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{37}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{35}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{33}\! \left(x \right)+F_{7}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{65}\! \left(x \right)
F_{38}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{64}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{57}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{33}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{41}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{41}\! \left(x \right)
F_{63}\! \left(x \right) = 0
F_{64}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{65}\! \left(x \right) = F_{19}\! \left(x \right)+F_{61}\! \left(x \right)+F_{66}\! \left(x \right)+F_{73}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{70}\! \left(x \right)
F_{69}\! \left(x \right) = F_{52}\! \left(x \right)
F_{70}\! \left(x \right) = F_{55}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{59}\! \left(x \right)+F_{66}\! \left(x \right)
F_{73}\! \left(x \right) = 0
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{2}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{2}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_17(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_74(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_37(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_1(x) + F_4(x))
Eq(F_26(x), F_19(x) + F_27(x) + F_35(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_30(x) + F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_11(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_4(x)*F_7(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_33(x) + F_7(x))
Eq(F_37(x), F_38(x) + F_65(x))
Eq(F_38(x), F_19(x) + F_20(x) + F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_19(x) + F_42(x) + F_64(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_46(x))
Eq(F_44(x), F_4(x) + F_45(x))
Eq(F_45(x), F_19(x) + F_27(x) + F_34(x))
Eq(F_46(x), F_41(x) + F_47(x))
Eq(F_47(x), F_19(x) + F_48(x) + F_61(x) + F_63(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_57(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_4(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_33(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), 2*F_19(x) + F_48(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_41(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_41(x))
Eq(F_63(x), 0)
Eq(F_64(x), F_2(x)*F_4(x))
Eq(F_65(x), F_19(x) + F_61(x) + F_66(x) + F_73(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_71(x))
Eq(F_68(x), F_69(x) + F_70(x))
Eq(F_69(x), F_52(x))
Eq(F_70(x), F_55(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), 2*F_19(x) + F_59(x) + F_66(x))
Eq(F_73(x), 0)
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_2(x) + F_77(x))
Eq(F_77(x), F_19(x) + F_42(x) + F_78(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_2(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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