0132_0213_1230_2130_3021
Counting sequence:
1, 1, 2, 6, 19, 52, 127, 296, 681, 1571, 3656, 8590, 20348, 48498, 116094, 278721, 670457, 1614808, 3892485, 9387804, 22649119, 54655855, 131912630, 318403922, 768596538, 1855397622, 4479070556, 10813020589, 26104275231, 63020219724, 152142530667, 367301750080, 886740320789, 2140773156235, 5168271694084, 12477292376838, 30122817348856, 72722863816778, 175568442635354, 423859583493257, 1023287341692229, 2470433833366608, 5964154306998641, 14398741312441172, 34761635095547979, 83922008532299319, 202605647352595186, 489133295458721386, 1180872225683740790, 2850877726461161774, 6882627645654752056, 16616132964454340373, 40114893488295824939, 96845919801462088868, 233806732865368527575, 564459385166763803928, 1362725502607609358977, 3289910389425260446515, 7942546279910121443904, 19175002946740772497214, 46292552169338926842228, 111760107278861155800418, 269812766716451028520630, 651385640694595532595985, 1572584048077864203605649, 3796553736805378369519112, 9165691521615897482316637, 22127936779919504303608396, 53421565081264513598737271, 128971066942140469979821631, 311363698965046999546405806, 751698464871427953539483970, 1814760628706597937080340658, 4381219722282512342622078118, 10577200073268206167701476244, 25535619868813396738324026941, 61648439810886055250025614647, 148832499490571034904350450748, 359313438792004708330377913795, 867459377074542562502732994224, 2094232192941028527545122144749, 5055923762956500422739882376475, 12206079718853868872381070372860, 29468083200663978472005111835702, 71142246120181405620250566384368, 171752575441026109820868605815482, 414647397002232525174209411730066, 1001047369445489380189871424371289, 2416742135893208405486757889459421, 5834531641231901531116776827555584, 14085805418357003927606506798012617, 34006142477945897186169375301485060, 82098090374248778559671037532532899, 198202323226443422365076815376216455, 478502736827135571609115813426400234, 1155207796880714481962164952380343258, 2788918330588564400231593373480081646, 6733044458057843063502355864785069918, 16255007246704250173011456923788030064, 39243058951466342836377425698543760677, 94741125149636934918393616127796257859
Generating function in Maple syntax:
(x^11+x^10-5*x^9-x^8+9*x^7+2*x^5-4*x^4-9*x^3+13*x^2-6*x+1)/(x^2+x-1)/(x^2+2*x-1)/(x-1)^4
Generating function in latex syntax:
\frac{x^{11}+x^{10}-5 x^{9}-x^{8}+9 x^{7}+2 x^{5}-4 x^{4}-9 x^{3}+13 x^{2}-6 x +1}{\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**11 + x**10 - 5*x**9 - x**8 + 9*x**7 + 2*x**5 - 4*x**4 - 9*x**3 + 13*x**2 - 6*x + 1)/((x - 1)**4*(x**2 + x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+x-1)*(x^2+2*x-1)*(x-1)^4*F(x)+x^11+x^10-5*x^9-x^8+9*x^7+2*x^5-4*x^4-9*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{11}+x^{10}-5 x^{9}-x^{8}+9 x^{7}+2 x^{5}-4 x^{4}-9 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
1/6*n^3-n^2+piecewise(Or(n = 0,n = 1),3,n = 2,2,n = 3,1,0)-1/6*n-5-2/5*(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+2/5*(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)+(-1/2*5^(1/2)-1/2)^(-n)+(1/2*5^(1/2)-1/2)^(-n)+1/2*(2^(1/2)-1)^(-n)+1/2*(-1-2^(1/2))^(-n)
Explicit closed form in latex syntax:
\frac{n^{3}}{6}-n^{2}+\left(\left\{\begin{array}{cc}3 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 1 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)-\frac{n}{6}-5-\frac{2 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}+\frac{2 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}+\left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+\frac{\left(\sqrt{2}-1\right)^{-n}}{2}+\frac{\left(-1-\sqrt{2}\right)^{-n}}{2}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 52
a(6) = 127
a(7) = 296
a(8) = 681
a(9) = 1571
a(10) = 3656
a(11) = 8590
a(n+1) = 1/9*n^3-n^2-1/3*a(n)+a(n+3)-1/3*a(n+4)-1/9*n-4/3, n >= 12
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) = 52
a \! \left(6\right) = 127
a \! \left(7\right) = 296
a \! \left(8\right) = 681
a \! \left(9\right) = 1571
a \! \left(10\right) = 3656
a \! \left(11\right) = 8590
a \! \left(n +1\right) = \frac{n^{3}}{9}-n^{2}-\frac{a \! \left(n \right)}{3}+a \! \left(n +3\right)-\frac{a \! \left(n +4\right)}{3}-\frac{n}{9}-\frac{4}{3}, \quad n \geq 12
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18488/
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[23,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[21,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[14,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[13,x]
F[23,x] = F[24,x]+F[34,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[24,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[24,x]
F[34,x] = F[35,x]+F[36,x]+F[66,x]
F[35,x] = 0
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[53,x]
F[38,x] = F[39,x]+F[6,x]
F[39,x] = F[35,x]+F[40,x]+F[45,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[10,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[28,x]+F[42,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[44,x]+F[47,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[12,x]
F[51,x] = F[52,x]
F[52,x] = F[12,x]*F[47,x]
F[53,x] = F[34,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[35,x]+F[58,x]+F[64,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[62,x]
F[60,x] = F[10,x]+F[61,x]
F[61,x] = F[35,x]+F[40,x]+F[43,x]
F[62,x] = F[57,x]+F[63,x]
F[63,x] = F[55,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[24,x]+F[57,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[65,x]+F[68,x]
F[68,x] = F[69,x]+F[72,x]
F[69,x] = F[70,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[28,x]
F[72,x] = 2*F[35,x]+F[73,x]+F[75,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[42,x]
F[75,x] = F[12,x]*F[68,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_{23}\! \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_{21}\! \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_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{34}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \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_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \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_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{24}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{66}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{53}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{6}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{28}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{53}\! \left(x \right) = F_{34}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{35}\! \left(x \right)+F_{58}\! \left(x \right)+F_{64}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{10}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{62}\! \left(x \right) = F_{57}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{55}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{24}\! \left(x \right)+F_{57}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{72}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{28}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{73}\! \left(x \right)+F_{75}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{42}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \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_23(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_21(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_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_14(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_13(x))
Eq(F_23(x), F_24(x) + F_34(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_1(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_24(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_24(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_66(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_53(x))
Eq(F_38(x), F_39(x) + F_6(x))
Eq(F_39(x), F_35(x) + F_40(x) + F_45(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_10(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_28(x) + F_42(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_44(x) + F_47(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_12(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_12(x)*F_47(x))
Eq(F_53(x), F_34(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_35(x) + F_58(x) + F_64(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_62(x))
Eq(F_60(x), F_10(x) + F_61(x))
Eq(F_61(x), F_35(x) + F_40(x) + F_43(x))
Eq(F_62(x), F_57(x) + F_63(x))
Eq(F_63(x), F_55(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_24(x) + F_57(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_65(x) + F_68(x))
Eq(F_68(x), F_69(x) + F_72(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_28(x))
Eq(F_72(x), 2*F_35(x) + F_73(x) + F_75(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_42(x))
Eq(F_75(x), F_12(x)*F_68(x))
Pack JSON:
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Specification JSON:
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