0,2

I.e., n^2 + {1 + 4 + 9 + 16 + ... + m^2} = a(n) = A065612(n)^2 = A065311(n). a(n) is the smallest square obtained as n^2 + x*(x+1)*(2x+1)/6 where x = A065610(n).

Harry J. Smith, Table of n, a(n) for n = 0..500

n = 3: a(3) = 64 because n^2 + 1 + 4 + 9 + 16 + 25 = 9 + (1 + 4 + 9 + 16 + 25) = 64 = 8^2;

n = 4: a(4) = 150700176 because n^2 + (1 + 4 + ... + 767^2) = 150700176 = 12276^2, where 767 is the length of the shortest such consecutive-square sequence which provides(when summed) a new square, namely 12276^2. Often the least solution is rather large. E.g., a(93) = 23850559947150225 which means that 93^2 + A000330(415151) = 8649 + [a long square sum] = 154436265^2 = 23850559947150225.

Do[s = n^2; k = 1; While[s = s + k^2; !IntegerQ[ Sqrt[s]], k++ ]; Print[s], {n, 0, 30} ]

(PARI) { for (n = 0, 500, s=n^2 + 1; k=1; while (!issquare(s), k++; s+=k^2); write("b065611.txt", n, " ", s) ) } \\ Harry J. Smith, Oct 23 2009

(PARI) a(n) = my(s=n^2+1, k=1); while (!issquare(s), k++; s+=k^2); s; \\ Michel Marcus, Mar 24 2020

Cf. A000330, A065610, A065612.

Sequence in context: A161023 A252920 A156405 * A101252 A133281 A326320

Adjacent sequences: A065608 A065609 A065610 * A065612 A065613 A065614

Labos Elemer, Nov 07 2001

Edited by Jon E. Schoenfield, Jun 14 2018

Name clarified by Michel Marcus, Mar 24 2020

approved