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A248698
Floor of sums of the non-integer fourth roots of n, as partitioned by the integer roots: floor(Sum_{j=n^4+1..(n+1)^4-1} j^(1/4)).
4
0, 23, 166, 621, 1676, 3715, 7218, 12761, 21016, 32751, 48830, 70213, 97956, 133211, 177226, 231345, 297008, 375751, 469206, 579101, 707260, 855603, 1026146, 1221001, 1442376, 1692575, 1973998, 2289141, 2640596, 3031051, 3463290, 3940193, 4464736, 5039991
OFFSET
0,2
COMMENTS
The fractional portion of each sum converges to 3/10.
See A247112 for references to other related sequences and a conjecture.
FORMULA
a(n) = floor(Sum_{j=n^4+1..(n+1)^4-1} j^(1/4)).
a(n) = 3*n + 8*n^2 + 8*n^3 + 4*n^4.
G.f.: -x*(x^3+21*x^2+51*x+23) / (x-1)^5. - Colin Barker, Dec 30 2014
E.g.f.: exp(x)*x*(23 + 60*x + 32*x^2 + 4*x^3). - Stefano Spezia, Jul 13 2024
MATHEMATICA
Table[3 n + 8 n^2 + 8 n^3 + 4 n^4, {n, 0, 50}]
PROG
(PARI) a(n) = floor(sum(j=n^4+1, (n+1)^4-1, j^(1/4))); \\ Michel Marcus, Dec 22 2014
(PARI) concat(0, Vec(-x*(x^3+21*x^2+51*x+23)/(x-1)^5 + O(x^100))) \\ Colin Barker, Dec 30 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Richard R. Forberg, Dec 02 2014
STATUS
approved