A213083 Each square n^2 appears n^2 number of times.

1, 4, 4, 4, 4, 9, 9, 9, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36

Offset: 1

Author

Jens Ahlström, Jun 05 2012

Keywords

• nonn

Crossrefs

Cf. A000290, A006752, A074279.

Programs

Mathematica

Flatten[Table[Table[n^2, {n^2}], {n, 6}]] (* T. D. Noe, Jun 05 2012 *)

Python

from sympy.core.intfunc import integer_nthroot
def A213083(n): return ((m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))**2 # Chai Wah Wu, Jun 21 2025

Formula

a(n) = A074279(n)^2. Alternatively, a(n) = (m+1)^2 if n > m(m+1)(2m+1)/6 and a(n) = m^2 otherwise where m = floor((3n)^(1/3)). - Chai Wah Wu, Jun 21 2025

Sum_{n>=1} (-1)^(n+1)/a(n) = A006752. - Amiram Eldar, Jun 30 2025