A087059 Difference between 2*n^2 and the next greater square number.

2, 1, 7, 4, 14, 9, 2, 16, 7, 25, 14, 1, 23, 8, 34, 17, 47, 28, 7, 41, 18, 56, 31, 4, 46, 17, 63, 32, 82, 49, 14, 68, 31, 89, 50, 9, 71, 28, 94, 49, 2, 72, 23, 97, 46, 124, 71, 16, 98, 41, 127, 68, 7, 97, 34, 128, 63, 161, 94, 25, 127, 56, 162, 89, 14, 124, 47, 161, 82, 1, 119

Offset

1,1

Comments

For n >= 2, a(n) is also the smallest absolute value of all negative values in row n of the triangle D(n, m) = n^2 - m^2 - 2*n*m, for 2 <= m + 1 <= n. The negative values in row n start with m = floor(n/(1 + sqrt(2))) + 1 = ceiling(n/(1 + sqrt(2))). See also a comment on A087056 for the smallest positive numbers in row n >= 3. - Wolfdieter Lang, Jun 11 2015

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

a(n) = A087058(n) - 2*n^2 = A087057(n)^2 - 2*n^2 = (1 + A001951(n))^2 - 2*n^2 = (1 + floor(n*sqrt(2)))^2 - 2*n^2.

a(n) = 2*c(n)^2 - (n - c(n))^2, with c(n) := ceiling(n/(1 + sqrt(2))), n >= 1. - Wolfdieter Lang, Jun 11 2015

Example

a(10) = 25 because the difference between 2*10^2 = 200 and the next greater square number (225) is 25.

Mathematica

(Floor[Sqrt[#]]+1)^2-#&/@Table[2n^2, {n, 80}] (* Harvey P. Dale, Jan 15 2023 *)

Prog

(PARI) a(n) = (1 + sqrtint(2*n^2))^2 - 2*n^2 \\ Michel Marcus, Jun 25 2013

Crossrefs

Cf. A001951, A087055, A087056, A087057, A087058, A087060.

Sequence in context: A075085 A217458 A124048 * A120872 A204771 A141512

Adjacent sequences: A087056 A087057 A087058 * A087060 A087061 A087062

Keyword

easy,nonn

Author

Jens Voß, Aug 07 2003

Status

approved