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

0,2

Comments

These coefficients are called the Tripell numbers by Bravo et al. - Rigoberto Florez, Jan 23 2020

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).

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

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

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 *)

Prog

(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 */
(PARI) Vec(1/(1-2*x-x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012
(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
(Sage)
def A077939_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return (1/(1-2*x-x^2-x^3)).list()
A077939_list(30) # G. C. Greubel, Feb 05 2020
(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

Crossrefs

Sequence in context: A007443 A120925 A086588 * A077986 A007020 A080888

Adjacent sequences: A077936 A077937 A077938 * A077940 A077941 A077942

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 17 2002

Extensions

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

Status

approved