OFFSET
0,3
COMMENTS
In general, sqrt( Sum_{n>=0} x^n/(1 - q^n*x) ) is an integer series whenever q == 1 (mod 4).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..75
FORMULA
G.f.: A(x) = sqrt( Sum_{n>=0} x^n/(1-5^n*x) ).
EXAMPLE
A(x) = 1 + x + 3*x^2 + 23*x^3 + 411*x^4 + 15771*x^5 + ...
A(x)^2 = 1 + 2*x + 7*x^2 + 52*x^3 + 877*x^4 + 32502*x^5 + ...
= 1/(1-x) + x/(1-5x) + x^2/(1-25x) + x^3/(1-125x) + ...
MATHEMATICA
With[{m = 30}, CoefficientList[Series[Sqrt[Sum[x^j/(1 - 5^j*x), {j, 0, m + 2}]], {x, 0, m}], x]] (* G. C. Greubel, Jun 30 2021 *)
PROG
(PARI) a(n)=polcoeff(sqrt(sum(k=0, n, sum(j=0, k, (5^j)^(k-j) )*x^k+x*O(x^n))), n)
(Magma)
m:=30;
R<x>:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( Sqrt( (&+[x^j/(1-5^j*x): j in [0..m+2]]) ) )); // G. C. Greubel, Jun 30 2021
(Sage)
m=30;
def A118195_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( sqrt(sum( x^j/(1-5^j*x) for j in (0..m+2))) ).list()
A118195_list(m) # G. C. Greubel, Jun 30 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 15 2006
STATUS
approved