A320278 a(n) is the number of positive integers 0 < i < n such that i + n is a square.

0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5

Offset

0,14

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

a(n) = floor(sqrt(max(0,2*n-1))) - floor(sqrt(n)). - Alois P. Heinz, Oct 27 2018

a(n) = Sum_{i=1..floor((2*n-1)/2)} c(2*n-i), where c is the square characteristic (A010052). - Wesley Ivan Hurt, Nov 26 2020

Example

13 + 3 is a square and 13 + 12 is a square, so a(13)=2.

Prog

(PARI) a(n) = sum(i=1, n-1, issquare(i+n)); \\ Michel Marcus, Oct 09 2018
(PARI) a(n) = sqrtint(n<<1-1) - sqrtint(n+1) + issquare(n+1) \\ David A. Corneth, Oct 09 2018

Crossrefs

Cf. A000196, A010052.

Sequence in context: A135265 A144110 A076490 * A302111 A124278 A253187

Adjacent sequences: A320275 A320276 A320277 * A320279 A320280 A320281

Keyword

nonn,easy

Author

Jud McCranie, Oct 09 2018

Extensions

Offset changed to 0 by David A. Corneth, Oct 09 2018

Status

approved