A053187 Square nearest to n.

0, 1, 1, 4, 4, 4, 4, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Author

Henry Bottomley, Mar 01 2000

Keywords

• easy
• nonn

Comments

Apart from 0, k^2 appears 2k times from a(k^2-k+1) through to a(k^2+k) inclusive.

Examples

			a(7) = 9 since 7 is closer to 9 than to 4.
G.f. = x + x^2 + 4*x^3 + 4*x^4 + 4*x^5 + 4*x^6 + 9*x^7 + 9*x^8 + 9*x^9 + ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A048760, A053188, A002483, A245552.

Cf. A061023, A201053 (nearest cube), A000290, A000194.

Programs

Haskell

a053187 n = a053187_list !! n
a053187_list = 0 : concatMap (\x -> replicate (2*x) (x ^ 2)) [1..]
-- Reinhard Zumkeller, Nov 28 2011

Maple

seq(ceil((-1+sqrt(4*n+1))/2)^2, n=0..20); # Robert Israel, Jan 05 2015

Mathematica

nearestSq[n_] := Block[{a = Floor@ Sqrt@ n}, If[a^2 + a + 1/2 > n, a^2, a^2 + 2 a + 1]]; Array[ nearestSq, 75, 0] (* Robert G. Wilson v, Aug 01 2014 *)

Python

from math import isqrt
def A053187(n): return ((m:=isqrt(n))+int(n>m*(m+1)))**2 # Chai Wah Wu, Jun 06 2025

Formula

a(n) = ceiling((-1 + sqrt(4*n+1))/2)^2. - Robert Israel, Aug 01 2014

G.f.: (1/(1-x))*Sum_{n>=0} (2*n+1)*x^(n^2+n+1). - Robert Israel, Aug 01 2014. This is related to the Jacobi theta-function theta'_1(q), see A002483 and A245552.

G.f.: x / (1-x) * Sum_{k>0} (2*k - 1) * x^(k^2 - k). - Michael Somos, Jan 05 2015

a(n) = floor(sqrt(n)+1/2)^2. - Mikael Aaltonen, Jan 17 2015

Sum_{n>=1} 1/a(n)^2 = 2*zeta(3). - Amiram Eldar, Aug 15 2022

a(n) = A000194(n)^2. - Chai Wah Wu, Jun 06 2025

Extensions

Title improved by Jon E. Schoenfield, Jun 09 2019