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

0,4

Comments

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

Links

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

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

Cf. A048760, A053188, A002483, A245552.

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

Sequence in context: A361471 A124570 A213083 * A013189 A295643 A190718

Adjacent sequences: A053184 A053185 A053186 * A053188 A053189 A053190

Keyword

easy,nonn

Author

Henry Bottomley, Mar 01 2000

Extensions

Title improved by Jon E. Schoenfield, Jun 09 2019

Status

approved