A373330 a(n) is the difference between T = A000217(n^2) and the greatest square not exceeding T.

0, 0, 1, 9, 15, 1, 41, 0, 55, 72, 9, 156, 36, 204, 262, 144, 135, 289, 209, 316, 111, 117, 406, 309, 527, 261, 342, 860, 804, 36, 954, 1200, 624, 605, 1257, 969, 1400, 741, 849, 1856, 1639, 0, 1721, 2076, 855, 701, 1770, 1101, 1719, 397, 426, 1980, 1416, 2449, 1142

Offset: 0

Author

Hugo Pfoertner, Jun 02 2024

Keywords

• nonn
• look

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..9998
• Hugo Pfoertner, Logarithmic plot of a(n) vs n, n <= 10^5, lower envelope of terms > 0 shown in red, zoom for details.

Crossrefs

Cf. A000217, A000290, A002315, A037270, A061288, A373329, A373333.

A373331 and A373332 are the coordinates of the observed lower envelope of this sequence.

Programs

Mathematica

Array[PolygonalNumber[#^2] - Floor[Sqrt[(#^4 + #^2)/2]]^2 &, 55, 0] (* Michael De Vlieger, Jun 02 2024 *)

PARI

a(n) = my(T=(n^4+n^2)/2); T-sqrtint(T)^2

Python

from sympy import integer_nthroot
def A373330(n): return (T:=(n**4 + n**2) // 2)-(integer_nthroot(T,2)[0])**2
# Karl-Heinz Hofmann, Jul 01 2024

Formula

a(n) = A000217(n^2) - A373329(n)^2.

a(A002315(n)) = 0.