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A363498
a(n) = Sum_{k=0..n} floor(sqrt(k))^4.
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
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
KEYWORD
nonn,easy
AUTHOR
Hans J. H. Tuenter, Jun 05 2023
STATUS
approved