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)).

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

Author

Richard R. Forberg, Dec 02 2014

Keywords

• nonn
• easy

Comments

The fractional portion of each sum converges to 3/10.

See A247112 for references to other related sequences and a conjecture.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A247112, A248575, A248621.

Programs

Mathematica

Table[3 n + 8 n^2 + 8 n^3 + 4 n^4, {n, 0, 50}]

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

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