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
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