A363498 - OEIS

%I #31 Jul 20 2023 20:17:09

%S 0,1,2,3,19,35,51,67,83,164,245,326,407,488,569,650,906,1162,1418,

%T 1674,1930,2186,2442,2698,2954,3579,4204,4829,5454,6079,6704,7329,

%U 7954,8579,9204,9829,11125,12421,13717,15013,16309,17605,18901,20197,21493,22789

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

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

%H Karl-Heinz Hofmann, <a href="/A363498/b363498.txt">Table of n, a(n) for n = 0..10000</a>

%F 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)).

%t 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 *)

%o (Python)

%o from math import isqrt

%o def A363498(n):

%o     return (m:=isqrt(n))**4 *(n+1) - (m*(m+1)*(20*m**4+4*m**3-14*m**2+4*m+1))//30

%o print([A363498(n) for n in range(0,46)]) # _Karl-Heinz Hofmann_, Jul 15 2023

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

%K nonn,easy

%O 0,3

%A _Hans J. H. Tuenter_, Jun 05 2023