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

0,3

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

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).

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

Example

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.

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 *)

Prog

(Magma) k:=1; f:=func<n | n*(k*n+1)>; [0] cat [f(n*m): m in [-1, 1], n in [1..30]]; // Bruno Berselli, Nov 14 2012
(PARI) a(n)=n\=2; n*(n+1) \\ Charles R Greathouse IV, Jul 05 2013

Crossrefs

Cf. A109613.

Partial sums give A006584.

Sequence in context: A028476 A179478 A051548 * A332759 A139550 A337735

Adjacent sequences: A110657 A110658 A110659 * A110661 A110662 A110663

Keyword

nonn,easy

Author

Reinhard Zumkeller, Aug 05 2005

Extensions

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

Status

approved