A077939 Expansion of 1/(1 - 2*x - x^2 - x^3).
1, 2, 5, 13, 33, 84, 214, 545, 1388, 3535, 9003, 22929, 58396, 148724, 378773, 964666, 2456829, 6257097, 15935689, 40585304, 103363394, 263247781, 670444260, 1707499695, 4348691431, 11075326817, 28206844760, 71837707768, 182957587113, 465959726754
Offset: 0
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..2463
- Jhon Jairo Bravo, Maribel Diaz, and José Luis Ramirez, The 2-adic and 3-adic valuation of the Tripell sequence and an application, Turk J Math, (2020) 44: 131-141.
- Jhon J. Bravo, Jose L. Herrera, and José L. Ramírez, Combinatorial Interpretation of Generalized Pell Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.2.1.
- Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
- Index entries for linear recurrences with constant coefficients, signature (2,1,1).
Programs
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GAP
a:=[1,2,5];; for n in [4..30] do a[n]:=2*a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Feb 05 2020
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Magma
I:=[1,2,5]; [n le 3 select I[n] else 2*Self(n-1) +Self(n-2) +Self(n-3): n in [1..30]]; // G. C. Greubel, Feb 05 2020
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Maple
m:=30; S:=series(1/(1-2*x-x^2-x^3), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 05 2020
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Mathematica
CoefficientList[Series[1/(1-2*x-x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{2,1,1},{1,2,5},40] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *) a[n_]:=a[n]=2a[n-1]+a[n-2]+a[n-3]; a[0]=0; a[1]=1; a[2]=2; Table[a[n], {n,30}] (* Rigoberto Florez, Jan 23 2020 *) -
Maxima
a(n):=sum(sum((sum(binomial(j,n-m-3*k+2*j+1)*binomial(k,j),j,0,k))* binomial(m+k-1,m-1),k,0,n-m+1),m,1,n+1); /* Vladimir Kruchinin, Oct 11 2011 */
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PARI
Vec(1/(1-2*x-x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012
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Sage
def A077939_list(prec): P.
= PowerSeriesRing(ZZ, prec) return (1/(1-2*x-x^2-x^3)).list() A077939_list(30) # G. C. Greubel, Feb 05 2020
Formula
a(n) = abs(A077986(n)) = A077849(n) - A077849(n-1) = |A077922(n)| + |A077922(n-1)| = Sum_{k=0..n} A077997(k). - Ralf Stephan, Feb 02 2004
a(n) = Sum_{m=1..n+1} Sum_{k=0..n-m+1} (Sum_{j=0..k} binomial(j,n-m-3*k+2*j+1) *binomial(k,j))*binomial(m+k-1,m-1). - Vladimir Kruchinin, Oct 11 2011
G.f. for sequence with 1 prepended: 1/(1 - Sum_{k>=0} x*(x+x^2+x^3)^k). - Joerg Arndt, Sep 30 2012
G.f.: Q(0)/2, where Q(k) = 1 + 1/(1- x*(4*k+2 + x+x^2)/(x*(4*k+4 + x+x^2) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 04 2013
a(n) = 2*a(n-1) + a(n-2) + a(n-3), where a(0) = 0, a(1)=1, a(2)=2. - Rigoberto Florez, Jan 23 2020
Extensions
Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021
Comments