A363498 a(n) = Sum_{k=0..n} floor(sqrt(k))^4.

0, 1, 2, 3, 19, 35, 51, 67, 83, 164, 245, 326, 407, 488, 569, 650, 906, 1162, 1418, 1674, 1930, 2186, 2442, 2698, 2954, 3579, 4204, 4829, 5454, 6079, 6704, 7329, 7954, 8579, 9204, 9829, 11125, 12421, 13717, 15013, 16309, 17605, 18901, 20197, 21493, 22789

Offset

0,3

Comments

Partial sums of the fourth powers of the terms of A000196.

Links

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Formula

a(n) = (n+1)*m^4 - (1/30)*m*(m+1)*(20*m^4+4*m^3-14*m^2+4*m+1), where m = floor(sqrt(n)).

Mathematica

Table[(n + 1) #^4 - (1/30) # (# + 1)*(20 #^4 + 4 #^3 - 14 #^2 + 4 # + 1) &[Floor@ Sqrt[n]], {n, 0, 45}] (* Michael De Vlieger, Jun 10 2023 *)

Prog

(Python)
from math import isqrt
def A363498(n):
    return (m:=isqrt(n))**4 *(n+1) - (m*(m+1)*(20*m**4+4*m**3-14*m**2+4*m+1))//30
print([A363498(n) for n in range(0, 46)]) # Karl-Heinz Hofmann, Jul 15 2023

Crossrefs

Sums of powers of A000196: A022554 (1st), A174060 (2nd), A363497 (3rd), this sequence (4th), A363499 (5th).

Sequence in context: A213896 A088790 A283186 * A215304 A215281 A215153

Adjacent sequences: A363495 A363496 A363497 * A363499 A363500 A363501

Keyword

nonn,easy

Author

Hans J. H. Tuenter, Jun 05 2023

Status

approved