A088132 a(n) equals the square of the n-th partial sum added to twice the n-th partial sum of the squares, divided by a(n-1), for all n>1, with a(0)=1, a(1)=3.

1, 3, 12, 47, 185, 728, 2865, 11275, 44372, 174623, 687217, 2704496, 10643361, 41886227, 164840412, 648718287, 2552986921, 10047107272, 39539710801, 155605856283, 612376317860, 2409965560639, 9484256386273, 37324649227232

Offset: 0

Author

Paul D. Hanna, Sep 19 2003

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (4,0,-1).

Crossrefs

Cf. A030186, A088131.

Programs

GAP

a:=[1,3,12];; for n in [4..30] do a[n]:=34a[n-1]-a[n-3]; od; a; # G. C. Greubel, Oct 26 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-4*x+x^3) )); // G. C. Greubel, Oct 26 2019

Maple

seq(coeff(series((1-x)/(1-4*x+x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 26 2019

Mathematica

LinearRecurrence[{4,0,-1}, {1,3,12}, 30] (* or *) CoefficientList[Series[ (1-x)/(1-4x+x^3), {x,0,30}], x] (* Harvey P. Dale, Jun 24 2011 *)

PARI

{a(n)=if(n==0,1, if(n==1,3, (sum(k=0, n-1, a(k))^2 + 2*sum(k=0, n-1, a(k)^2))/a(n-1)))}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Feb 20 2014

PARI

Vec( (1-x)/(1-4*x+x^3) + O(x^66) ) \\ Joerg Arndt, Feb 16 2014

Sage

def A088132_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-4*x+x^3)).list()
A088132_list(30) # G. C. Greubel, Oct 26 2019

Formula

a(n) = 4*a(n-1) - a(n-3) for n>3.

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

G.f.: 1/(x - x^2*Sum_{n>=0} A030186(n)*x^n) - 1/x.