|
| |
|
|
A087059
|
|
Difference between 2*n^2 and the next greater square number.
|
|
10
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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
|
|
|
|
KEYWORD
|
easy,nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
