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A363497
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a(n) = Sum_{k=0..n} floor(sqrt(k))^3.
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4
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0, 1, 2, 3, 11, 19, 27, 35, 43, 70, 97, 124, 151, 178, 205, 232, 296, 360, 424, 488, 552, 616, 680, 744, 808, 933, 1058, 1183, 1308, 1433, 1558, 1683, 1808, 1933, 2058, 2183, 2399, 2615, 2831, 3047, 3263, 3479, 3695, 3911, 4127, 4343, 4559, 4775, 4991, 5334
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OFFSET
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0,3
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COMMENTS
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Partial sums of the third powers of the terms of A000196.
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LINKS
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FORMULA
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a(n) = (n+1)*m^3 - (1/60)*m*(m+1)*(3*m-1)*(12*m^2+7*m-4), where m = floor(sqrt(n)).
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MATHEMATICA
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Table[(n + 1) #^3 - (1/60) # (# + 1) (3 # - 1) (12 #^2 + 7 # - 4) &[Floor@ Sqrt[n]], {n, 0, 50}] (* Michael De Vlieger, Jun 10 2023 *)
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PROG
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(PARI) a(n) = sum(k=0, n, sqrtint(k)^3); \\ Michel Marcus, Jun 06 2023
(Python)
from math import isqrt
(Python)
from math import isqrt
def A363497(n):return (m:=isqrt(n))**3*(n+1)-(m*(m+1)*(3*m-1)*(12*m**2+7*m-4))//60
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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