Review A005148

[David Broadhurst]

A005148 is based on a fascinating paper by Newman and Shanks,
with an even more interesting appendix by Don Zagier.
If one studies the appendix carefully, one eventually
sees that it leads to an efficient recursive algorithm
for the series itself, whereas the main paper (and at present
the EIS) treats each term in isolation, which is
enormously slower.

\\ Newman-Shanks numbers: Math Comp 42 (1984) 199
\\ Using Zagier's appendix one may compute 1000 terms of
\\ A005148 in 25 seconds running Pari-GP on a 500MHz Alpha.

{nt=1000;a=[1];b=[1];d=1;e=0;g=0;print(1);for(n=2,nt,
c=48*(a[n-1]+g)+128*(d-32*e);e=d;d=c;i=(n-1)\2;
g=12*if(n%2==0,a[n/2]^2)+24*sum(j=1,i,a[j]*a[n-j]);
h=12*if(n%2==0,b[n/2]^2)+24*sum(j=1,i,b[j]*b[n-j]);
f=(c+5*h)/n^2-g;a=concat(a,f);b=concat(b,n*f);print(f))}

David Broadhurst             Email:  D.Broadhurst at open.ac.uk
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