A335639 Sum of the positive differences of the cubed parts in each partition of n into two parts.

0, 0, 7, 26, 82, 180, 369, 648, 1096, 1700, 2575, 3690, 5202, 7056, 9457, 12320, 15904, 20088, 25191, 31050, 38050, 45980, 55297, 65736, 77832, 91260, 106639, 123578, 142786, 163800, 187425, 213120, 241792, 272816, 307207, 344250, 385074, 428868, 476881, 528200

Offset

1,3

Links

Table of n, a(n) for n=1..40.

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} (n-i)^3 - i^3.

From Stefano Spezia, Oct 03 2020: (Start)

G.f.: x^3*(7 + 12*x + 16*x^2 + 6*x^3 + x^4)/((1 - x)^5*(1 + x)^3).

a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n > 8.

(End)

Example

a(5) = 82; 5 has two partitions into two parts, (4,1) and (3,2). The sum of the positive differences of the cubed parts is then (4^3-1^3) + (3^3-2^3) =  63 + 19 = 82.

Mathematica

Table[Sum[(n - i)^3 - i^3, {i, Floor[(n-1)/2]}], {n, 50}]

Crossrefs

Sequence in context: A247557 A027937 A135026 * A027138 A282703 A240256

Adjacent sequences: A335636 A335637 A335638 * A335640 A335641 A335642

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 03 2020

Status

approved