0132_0213_1320_2301
Counting sequence:
1, 1, 2, 6, 20, 62, 172, 440, 1074, 2569, 6133, 14771, 36062, 89309, 224019, 567808, 1450867, 3729918, 9632543, 24960070, 64839613, 168752729, 439820196, 1147529385, 2996434989, 7829140621, 20465756446, 53517717614, 139986636198, 366240747118, 958332811062, 2507952211500, 6563913011156, 17180565361113, 44971340345063, 117720570449767, 308164600825816, 806721691986717, 2111897395419917, 5528764335269844, 14473983292874117, 37892360908885358, 99201450165492865, 259708691051750714, 679918025918910183, 1780032192564228505, 4660152163493341870, 12200371521356134013, 31940856847457085647, 83621987914780687517, 218924684684417796816, 573151221713540226754, 1500527291606340025600, 3928427275405756457742, 10284747779211485134832, 26925802551429813362176, 70492632853480186914790, 184552041965815214755241, 483163384957574395883241, 1264937896734125854110843, 3311649872899238933426946, 8670010857272462485092637, 22698380969535891605116759, 59425128592570698502695368, 155576997890647176254270727, 407305851244312774969701214, 1066340528172175038081576819, 2791715677931980118136725262, 7308806394943300874060873921, 19134703285536993619520047641, 50095303018945822215457936520, 131151204885856757488782274033, 343358309867737019174757948769, 898923721175579437883242651821, 2353412846575451570170486087826, 6161314804383675824019262721910, 16130531538241377004669412331914, 42230279753672057395553211962990, 110560307609437999593118717328890, 289450642847968350206059947240692, 757791620481119868669575159850600, 1983924217688696891091693625672441, 5193981030771582075183561927835003, 13598018870999271875615104604692847, 35600075574972678633973976807147996, 93202207839411654190931275687489373, 244006547914248064268068750027559793, 638817435845304099271772773972358236, 1672445759505607354864245171078561145, 4378519842439404207954953937678246990, 11463113767348377754268599038825074549, 30010821458677274025386807972576144962, 78569350606826534262963754466505513163, 205697230358088508645648314602191067225, 538522340460011351438268907690618069170, 1409869791007090265197733845170815726469, 3691087032531548883212083501224189022563, 9663391306528135262552818405306528950589, 25299086886934014660674975208305014011364, 66233869354036224231929314206827809728266, 173402521174699289060027381386617076203204
Generating function in Maple syntax:
(2*x^10-10*x^9+12*x^8+19*x^7-85*x^6+143*x^5-151*x^4+103*x^3-43*x^2+10*x-1)/(2*x-1)/(x^2-3*x+1)/(x-1)^6
Generating function in latex syntax:
\frac{2 x^{10}-10 x^{9}+12 x^{8}+19 x^{7}-85 x^{6}+143 x^{5}-151 x^{4}+103 x^{3}-43 x^{2}+10 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{6}}
Generating function in sympy syntax:
(2*x**10 - 10*x**9 + 12*x**8 + 19*x**7 - 85*x**6 + 143*x**5 - 151*x**4 + 103*x**3 - 43*x**2 + 10*x - 1)/((x - 1)**6*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x^2-3*x+1)*(x-1)^6*F(x)+2*x^10-10*x^9+12*x^8+19*x^7-85*x^6+143*x^5-151*x^4+103*x^3-43*x^2+10*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{6} F \! \left(x \right)+2 x^{10}-10 x^{9}+12 x^{8}+19 x^{7}-85 x^{6}+143 x^{5}-151 x^{4}+103 x^{3}-43 x^{2}+10 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/120*(-12*5^(1/2)+60)*(3/2-1/2*5^(1/2))^(-n)+1/120*(12*5^(1/2)+60)*(3/2+1/2*5^(1/2))^(-n)+1/120*n^5-1/24*n^4+5/24*n^3-59/24*n^2+257/60*n+3*2^(-1+n)-6)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-12 \sqrt{5}+60\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(12 \sqrt{5}+60\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{n^{5}}{120}-\frac{n^{4}}{24}+\\\frac{5 n^{3}}{24}-\frac{59 n^{2}}{24}+\frac{257 n}{60}+3 \,2^{-1+n}-6 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(6) = 172
a(7) = 440
a(8) = 1074
a(9) = 2569
a(10) = 6133
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)+1/120*(n-2)*(n-9)*(n^3+6*n^2+33*n+20), n >= 11
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) = 62
a \! \left(6\right) = 172
a \! \left(7\right) = 440
a \! \left(8\right) = 1074
a \! \left(9\right) = 2569
a \! \left(10\right) = 6133
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{\left(n -2\right) \left(n -9\right) \left(n^{3}+6 n^{2}+33 n +20\right)}{120}, \quad n \geq 11
Specification 1
Strategy pack name: requirement_placements
Tree: http://permpal.com/tree/21198/
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[13,x]*F[4,x]
F[4,x] = F[5,x]+F[59,x]
F[5,x] = F[22,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[13,x]*F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]
F[13,x] = x
F[14,x] = F[15,x]+F[7,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[17,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[13,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[16,x]
F[22,x] = F[18,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[27,x]
F[24,x] = 0
F[25,x] = F[13,x]*F[26,x]
F[26,x] = F[23,x]+F[7,x]
F[27,x] = F[13,x]*F[28,x]
F[28,x] = F[29,x]+F[34,x]
F[29,x] = F[18,x]+F[30,x]
F[30,x] = F[24,x]+F[31,x]+F[33,x]
F[31,x] = F[13,x]*F[32,x]
F[32,x] = F[11,x]+F[30,x]
F[33,x] = F[13,x]*F[29,x]
F[34,x] = F[35,x]+F[46,x]
F[35,x] = F[24,x]+F[31,x]+F[36,x]
F[36,x] = F[13,x]*F[37,x]
F[37,x] = F[29,x]+F[38,x]
F[38,x] = F[35,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[13,x]*F[41,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[43,x]
F[43,x] = F[13,x]*F[44,x]
F[44,x] = F[18,x]+F[42,x]
F[45,x] = F[40,x]
F[46,x] = 2*F[24,x]+F[40,x]+F[47,x]
F[47,x] = F[13,x]*F[48,x]
F[48,x] = F[49,x]+F[53,x]
F[49,x] = F[50,x]
F[50,x] = F[13,x]*F[51,x]
F[51,x] = F[13,x]+F[52,x]
F[52,x] = F[50,x]
F[53,x] = 2*F[24,x]+F[47,x]+F[54,x]
F[54,x] = F[13,x]*F[55,x]
F[55,x] = F[56,x]+F[58,x]
F[56,x] = F[57,x]
F[57,x] = F[13,x]*F[18,x]
F[58,x] = F[54,x]
F[59,x] = F[60,x]
F[60,x] = F[13,x]*F[20,x]*F[61,x]
F[61,x] = F[62,x]+F[72,x]
F[62,x] = F[63,x]*F[64,x]
F[63,x] = F[10,x]+F[29,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[18,x]*F[20,x]
F[66,x] = F[67,x]
F[67,x] = F[13,x]*F[18,x]*F[68,x]
F[68,x] = F[20,x]+F[69,x]+F[70,x]
F[69,x] = F[13,x]*F[68,x]
F[70,x] = F[71,x]
F[71,x] = F[13,x]*F[20,x]*F[68,x]
F[72,x] = F[18,x]*F[20,x]*F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[20,x]^2*F[11,x]*F[13,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)+F_{59}\! \left(x \right)
F_{5}\! \left(x \right) = F_{22}\! \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_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \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_{13}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{7}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{13}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{46}\! \left(x \right)
F_{35}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{13}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{29}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{35}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{40}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{40}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{53}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{47}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{13}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{72}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right) F_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{10}\! \left(x \right)+F_{29}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{18}\! \left(x \right) F_{20}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{20}\! \left(x \right)+F_{69}\! \left(x \right)+F_{70}\! \left(x \right)
F_{69}\! \left(x \right) = F_{13}\! \left(x \right) F_{68}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right) F_{68}\! \left(x \right)
F_{72}\! \left(x \right) = F_{18}\! \left(x \right) F_{20}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{20} \left(x \right)^{2} F_{11}\! \left(x \right) F_{13}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_5(x) + F_59(x))
Eq(F_5(x), F_22(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_13(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_14(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_13(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_7(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_13(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_16(x))
Eq(F_22(x), F_18(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_27(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_13(x)*F_26(x))
Eq(F_26(x), F_23(x) + F_7(x))
Eq(F_27(x), F_13(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_34(x))
Eq(F_29(x), F_18(x) + F_30(x))
Eq(F_30(x), F_24(x) + F_31(x) + F_33(x))
Eq(F_31(x), F_13(x)*F_32(x))
Eq(F_32(x), F_11(x) + F_30(x))
Eq(F_33(x), F_13(x)*F_29(x))
Eq(F_34(x), F_35(x) + F_46(x))
Eq(F_35(x), F_24(x) + F_31(x) + F_36(x))
Eq(F_36(x), F_13(x)*F_37(x))
Eq(F_37(x), F_29(x) + F_38(x))
Eq(F_38(x), F_35(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_13(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_13(x)*F_44(x))
Eq(F_44(x), F_18(x) + F_42(x))
Eq(F_45(x), F_40(x))
Eq(F_46(x), 2*F_24(x) + F_40(x) + F_47(x))
Eq(F_47(x), F_13(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_53(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_13(x)*F_51(x))
Eq(F_51(x), F_13(x) + F_52(x))
Eq(F_52(x), F_50(x))
Eq(F_53(x), 2*F_24(x) + F_47(x) + F_54(x))
Eq(F_54(x), F_13(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_58(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_13(x)*F_18(x))
Eq(F_58(x), F_54(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_13(x)*F_20(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_72(x))
Eq(F_62(x), F_63(x)*F_64(x))
Eq(F_63(x), F_10(x) + F_29(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_18(x)*F_20(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_13(x)*F_18(x)*F_68(x))
Eq(F_68(x), F_20(x) + F_69(x) + F_70(x))
Eq(F_69(x), F_13(x)*F_68(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_13(x)*F_20(x)*F_68(x))
Eq(F_72(x), F_18(x)*F_20(x)*F_73(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_11(x)*F_13(x)*F_20(x)**2)
Pack JSON:
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Specification JSON:
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