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%I #15 May 09 2014 22:58:42
%S 1,5,13,25,50,61,85,130,145,181,250,265,338,410,421,481,610,650,685,
%T 850,841,925,1183,1105,1250,1450,1405,1586,1810,1741,1861,2275,2210,
%U 2245,2650,2521,2665,3255,3050,3121,3731,3445,3770,4210,3961,4250,4810
%N The sum of a^2 + b^2 for all nonnegative integers a,b such that b^2 - a^2 = 2*n+1.
%C A sample of 54 terms found none with last digit 2,4,7, or 9, and both ending digit 0 and 5 had 17; 15 had final digit 1.
%C The square of 29 = a(20); a(47)-a(48)=1, probably the only time this will occur.
%C Eleven primes all ending in 1 were found.
%F For each pair of divisors d and d' of 2*n+1 with d*d'=2*n+1 and d<=d', find a and b to satisfy b-a=d and b+a=d', then compute a^2 + b^2. Find the sum of all these results.
%F If 2*n+1 is not a square, a(n)=sum[d(2*n+1)^2 {d(2*n+1) a divisor of 2*n+1}].
%F If 2*n+1 is a square, a(n)=(sum[d(2*n+1)^2 {d(2*n+1) a divisor of 2*n+1}] +
%F 2*n+1)/2.
%e For n=31, 2*31+1=63=3^2*7, with divisors 1,3,7,9,21,63.
%e Grouping in pairs 1*63=(b-a)*(b+a) gives a=31 and b=32; 3*21=(b-a)*(b+a) gives a=9 and b=12; 7*9=(b-a)*(b+a) gives a=1 and b=8.
%e The sum 1^2 + 8^2 + 9^2 + 12^2 + 31^2 + 32^2 = 2275 = a(31).
%o (PARI) a(n)=my(b,N=2*n+1);sum(a=0,n,if(issquare(N+a^2,&b),a^2+b^2)) \\ _Charles R Greathouse IV_, Apr 28 2014
%Y Cf. A237626.
%K nonn
%O 0,2
%A _J. M. Bergot_, Apr 26 2014
%E a(4) corrected by _Charles R Greathouse IV_, Apr 28 2014
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