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A077939 Expansion of 1/(1 - 2*x - x^2 - x^3). 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Sequence is called the Tripell numbers in Bravo et al. article. - 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, José L. Ramírez, Combinatorial Interpretation of Generalized Pell Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.2.1.

Yüksel Soykan, On Generalized Third-Order Pell Numbers, Asian Journal of Advanced Research and Reports (2019) Vol. 6, No. 1, Article No. AJARR.51635, 1-18.

Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.

Yüksel Soykan, Generalized Tribonacci Numbers: Summing Formulas, Int. J. Adv. Appl. Math. and Mech. (2020) Vol. 7, No. 3, 57-76.

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,changed

AUTHOR

N. J. A. Sloane, Nov 17 2002

STATUS

approved

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Last modified July 8 07:34 EDT 2020. Contains 335513 sequences. (Running on oeis4.)