A087056 Difference between 2 * n^2 and the next smaller square number.

1, 4, 2, 7, 1, 8, 17, 7, 18, 4, 17, 32, 14, 31, 9, 28, 2, 23, 46, 16, 41, 7, 34, 63, 25, 56, 14, 47, 1, 36, 73, 23, 62, 8, 49, 92, 34, 79, 17, 64, 113, 47, 98, 28, 81, 7, 62, 119, 41, 100, 18, 79, 142, 56, 121, 31, 98, 4, 73, 144, 46, 119, 17, 92, 169, 63, 142, 32, 113, 196, 82

Offset

1,2

Comments

The difference x - y between the legs of primitive Pythagorean triangles x^2 + y^2 = z^2 with even y is D(n, m) = n^2 - m^2 - 2*n*m (see A249866 for the restrictions on n and m to have primitive triangles, which are not used here except for 1 < = m <= n-1). Here a(n) is for positive D values the smallest number in row n, namely D(n, floor(n/(1 + sqrt(2)))), for n >= 3. For the smallest value |D| for negative D in row n >= 2 see A087059. - Wolfdieter Lang, Jun 11 2015

Links

Table of n, a(n) for n=1..71.

Formula

a(n) = 2*n^2 - A087055(n) = 2*n^2 - A001951(n)^2 = 2*n^2 - (floor[n*sqrt(2)])^2

a(n) = (n - f(n))^2 - 2*f(n)^2 with f(n) = floor(n/(1 + sqrt(2))), for n >= 1 (the values for n = 1, 2 have here been included). See comment above. - Wolfdieter Lang, Jun 11 2015

Example

a(10) = 4 because the difference between 2*10^2 = 200 and the next smaller square number (196) is 4.

Prog

(PARI) a(n) = 2*n^2 - sqrtint(2*n^2)^2; \\ Michel Marcus, Jul 08 2020

Crossrefs

Cf. A001951, A087055, A087057, A087058, A087059, A087060.

Sequence in context: A353575 A372766 A286842 * A076129 A260590 A010648

Adjacent sequences: A087053 A087054 A087055 * A087057 A087058 A087059

Keyword

easy,nonn,changed

Author

Jens Voß, Aug 07 2003

Status

approved