A248698 Floor of sums of the non-integer fourth roots of n, as partitioned by the integer roots: floor[sum(j from n^4+1 to (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

0,2

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).

Formula

a(n) = floor[sum(j from n^4+1 to (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

Mathematica

Table[AccountingForm[N[Sum[j^(1/4), {j, n^4 + 1, (n + 1)^4 - 1}], 20]], {n, 0, 50}]
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

Cf. A247112.

Sequence in context: A142091 A168027 A155842 * A186260 A229426 A274587

Adjacent sequences: A248695 A248696 A248697 * A248699 A248700 A248701

Keyword

nonn,easy

Author

Richard R. Forberg, Dec 02 2014

Status

approved