A103142 a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4), with a(0)=1, a(1)=2, a(3)=5, a(4)=13.

1, 2, 5, 13, 34, 88, 228, 591, 1532, 3971, 10293, 26680, 69156, 179256, 464641, 1204374, 3121801, 8091873, 20974562, 54367172, 140922580, 365278767, 946821848, 2454212215, 6361447625, 16489208080, 42740897848, 110786663616, 287164880785, 744346531114

Offset: 0

Author

Paul Barry, Jan 24 2005

Keywords

• easy
• nonn

Comments

Row sums of generalized Pascal matrix A103141.

Generalized Pell numbers.

Row sums of the tetranacci convolution triangle A202193. - Philippe Deléham, Feb 16 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
• Index entries for linear recurrences with constant coefficients, signature (2,1,1,1).

Crossrefs

Cf. A000129, A077939.

Row sums of A103141 and of A202193.

Programs

GAP

a:=[1,2,5,13];; for n in [5..40] do a[n]:=2*a[n-1]+a[n-2]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Feb 12 2020

Magma

I:=[1,2,5,13]; [n le 4 select I[n] else 2*Self(n-1)+Self(n-2)+Self(n-3) +Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 05 2012

Maple

m:=40; S:=series(1/(1-2*x-x^2-x^3-x^4), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 12 2020

Mathematica

LinearRecurrence[{2,1,1,1}, {1,2,5,13}, 40] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)

PARI

Vec(1/(1-2*x-x^2-x^3-x^4)+O(x^40)) \\ Charles R Greathouse IV, Jun 20 2011

Sage

def A103142_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-x^2-x^3-x^4) ).list()
A103142_list(40) # G. C. Greubel, Feb 12 2020

Formula

a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4).

G.f.: 1/(1 - 2*x - x^2 - x^3 - x^4).

Extensions

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