A110660 Oblong (promic) numbers repeated.

0, 0, 2, 2, 6, 6, 12, 12, 20, 20, 30, 30, 42, 42, 56, 56, 72, 72, 90, 90, 110, 110, 132, 132, 156, 156, 182, 182, 210, 210, 240, 240, 272, 272, 306, 306, 342, 342, 380, 380, 420, 420, 462, 462, 506, 506, 552, 552, 600, 600, 650, 650, 702, 702, 756, 756, 812, 812

Offset: 0

Author

Reinhard Zumkeller, Aug 05 2005

Keywords

• nonn
• easy

Comments

a(floor(n/2)) = A002378(n).

Sum of the even numbers among the smallest parts in the partitions of 2n into two parts (see example). - Wesley Ivan Hurt, Jul 25 2014

For n > 0, a(n-1) is the sum of the smallest parts of the partitions of 2n into two distinct even parts. - Wesley Ivan Hurt, Dec 06 2017

Examples

			a(4) = 6; The partitions of 2*4 = 8 into two parts are: (7,1), (6,2), (5,3), (4,4). The sum of the even numbers from the smallest parts of these partitions gives: 2 + 4 = 6.
a(5) = 6; The partitions of 2*5 = 10 into two parts are: (9,1), (8,2), (7,3), (6,4), (5,5). The sum of the even numbers from the smallest parts of these partitions gives: 2 + 4 = 6.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Eric Weisstein's World of Mathematics, Pronic Number
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A109613.

Partial sums give A006584.

Programs

Magma

k:=1; f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..30]]; // Bruno Berselli, Nov 14 2012

Maple

A110660:=n->floor(n/2)*(floor(n/2)+1): seq(A110660(n), n=0..50); # Wesley Ivan Hurt, Jul 25 2014

Mathematica

Table[Floor[n/2] (Floor[n/2] + 1), {n, 0, 50}] (* Wesley Ivan Hurt, Jul 25 2014 *)
CoefficientList[Series[2*x^2/((1 - x)^3*(1 + x)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jul 25 2014 *)
LinearRecurrence[{1,2,-2,-1,1},{0,0,2,2,6},60] (* Harvey P. Dale, Jan 23 2021 *)

PARI

a(n)=n\=2;n*(n+1) \\ Charles R Greathouse IV, Jul 05 2013

Formula

a(n) = floor(n/2) * (floor(n/2)+1).

a(n) = A028242(n) * A110654(n).

a(n) = A008805(n-2)*2, with A008805(-2) = A008805(-1) = 0.

From Wesley Ivan Hurt, Jul 25 2014: (Start)

G.f.: 2*x^2/((1-x)^3*(1+x)^2);

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), for n > 4;

a(n) = (2*n^2 + 2*n - 1 + (2*n + 1)*(-1)^n)/8. (End)

a(n) = Sum_{i=1..n; i even} i. - Olivier Pirson, Nov 05 2017

Extensions

Typo in description (Name) fixed by Harvey P. Dale, Jul 09 2021