A124400 a(n) = a(n-1) + 3*a(n-2) - a(n-4), with a(0)=1, a(1)=1, a(2)=4, a(3)=7.

1, 1, 4, 7, 18, 38, 88, 195, 441, 988, 2223, 4992, 11220, 25208, 56645, 127277, 285992, 642615, 1443946, 3244514, 7290360, 16381287, 36808421, 82707768, 185842671, 417584688, 938304280, 2108350576, 4737420745, 10644887785, 23918845740

Offset: 0

Author

Philippe Deléham, Dec 14 2006

Keywords

• easy
• nonn

Comments

Unsigned version of A077920.

The sequence is the INVERT transform of the aerated even-indexed Fibonacci numbers (i.e., of (1, 0, 3, 0, 8, 0, ...)). Sequence A131322 is the INVERT transform of the aerated odd-indexed Fibonacci numbers. - Gary W. Adamson, Feb 07 2014

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,0,-1).

Crossrefs

Cf. A131322.

Programs

GAP

a:=[1,1,4,7];; for n in [5..35] do a[n]:=a[n-1]+3*a[n-2]-a[n-4]; od; a; # G. C. Greubel, Dec 25 2019

Magma

I:=[1,1,4,7]; [n le 2 select I[n] else Self(n-1) +3*Self(n-2) -Self(n-4): n in [1..35]]; // G. C. Greubel, Dec 25 2019

Maple

seq(coeff(series(1/(1-x-3*x^2+x^4), x, n+1), x, n), n = 0..35); # G. C. Greubel, Dec 25 2019

Mathematica

LinearRecurrence[{1,3,0,-1}, {1,1,4,7}, 35] (* G. C. Greubel, Dec 25 2019 *)
CoefficientList[Series[1/(1-x-3x^2+x^4),{x,0,30}],x] (* Harvey P. Dale, Feb 01 2022 *)

PARI

my(x='x+O('x^35)); Vec(1/(1-x-3*x^2+x^4)) \\ G. C. Greubel, Dec 25 2019

Sage

def A124400_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x-3*x^2+x^4) ).list()
A124400_list(35) # G. C. Greubel, Dec 25 2019

Formula

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