A273190 a(n) is the number of nonnegative m < n for which m + n is a perfect square.

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

Offset

0,10

Links

Peter Kagey, Table of n, a(n) for n = 0..10000

Formula

a(n) = floor(sqrt(2*n-1))-floor(sqrt(n-1)) for n > 0. - Chai Wah Wu, May 25 2016

a(n) = A103128(n) - A000196(n-1); after previous formula. - Michel Marcus, May 25 2016

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

Example

a(1) = 1 because 1 + 0 is a perfect square.
a(2) = 0 because neither 2 + 0 nor 2 + 1 are perfect squares.
a(5) = 1 because 5 + 4 is a perfect square.
a(9) = 2 because 9 + 0 and 9 + 7 are perfect squares.

Mathematica

Table[Count[Range[0, n - 1], m_ /; IntegerQ@ Sqrt[m + n]], {n, 0, 120}] (* Michael De Vlieger, May 18 2016 *)

Prog

(Java)
int n = 100;
int[] terms = new int[n];
for (int i = 0; i < n; i++) {
     for (int j = 0; j < i; j++) {
          if (Math.sqrt(i+j) == Math.floor(Math.sqrt(i+j))) {
               terms[i]++;
          }
     }
     System.out.print(terms[i] + ", ");
}
(PARI) a(n) = sum(k=0, n-1, issquare(n+k)); \\ Michel Marcus, May 18 2016
(Haskell) a273190 n = length $ filter (>=n) $ takeWhile (< 2 * n) $ map (^2) [1..] -- Peter Kagey, May 25 2016
(Python)
from gmpy2 import isqrt
def A273190(n):
    return isqrt(2*n-1)-isqrt(n-1) if n > 0 else 0 # Chai Wah Wu, May 25 2016

Crossrefs

Cf. A273185, A273191.

Sequence in context: A072530 A184341 A090455 * A086288 A104360 A318998

Adjacent sequences: A273187 A273188 A273189 * A273191 A273192 A273193

Keyword

easy,nonn

Author

Alec Jones, May 17 2016

Status

approved