A038714 Pronic numbers repeated 4 times; a(n) = floor(n/4) * ceiling((n+1)/4).

0, 0, 0, 0, 2, 2, 2, 2, 6, 6, 6, 6, 12, 12, 12, 12, 20, 20, 20, 20, 30, 30, 30, 30, 42, 42, 42, 42, 56, 56, 56, 56, 72, 72, 72, 72, 90, 90, 90, 90, 110, 110, 110, 110, 132, 132, 132, 132, 156, 156, 156, 156, 182, 182, 182, 182, 210, 210, 210, 210, 240

Offset

0,5

Comments

From Wesley Ivan Hurt, Nov 25 2017: (Start)

a(n) is the sum of the smallest even parts in the partitions of n into two parts. For example, a(8) = 6; the partitions of 8 into two parts is (7,1), (6,2), (5,3) and (4,4). The sum of the smallest even parts is then 2+4 = 6.

For n>0, a(n-1) is the sum of the smallest even parts in the partitions of n into two distinct parts. For example, a(11) = 6; the partitions of 12 into two distinct parts is (11,1), (10,2), (9,3), (8,4) and (7,5). The sum of the smallest even parts is then 2+4 = 6. (End)

Links

Colin Barker, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9). - R. J. Mathar, Mar 11 2012

From Wesley Ivan Hurt, Nov 25 2017: (Start)

a(n) = floor(n/4) * (floor(n/4) + 1).

a(n) = Sum_{i=1..floor(n/2)} i * ((i+1) mod 2).

(End)

G.f.: 2*x^4 / ((1 - x)^3*(1 + x)^2*(1 + x^2)^2). - Colin Barker, Nov 26 2017

a(n) = A002378(A004526(n)). - Wesley Ivan Hurt, Nov 26 2017

a(n) = (2*n + 2*(-1)^((2*n + (-1)^n - 1)/4) + (-1)^n - 3)*(2*n + 2*(-1)^((2*n + (-1)^n - 1)/4) + (-1)^n + 5)/64. - Iain Fox, Nov 27 2017

Maple

A038714:=n->floor(n/4)*ceil((n+1)/4): seq(A038714(n), n=0..100); # Wesley Ivan Hurt, Nov 26 2017

Mathematica

Table[Floor[n/4] Ceiling[(n + 1)/4], {n, 0, 100}] (* Wesley Ivan Hurt, Nov 26 2017 *)
LinearRecurrence[{1, 0, 0, 2, -2, 0, 0, -1, 1}, {0, 0, 0, 0, 2, 2, 2, 2, 6}, 70] (* Harvey P. Dale, Jun 12 2022 *)

Prog

(PARI) concat(vector(4), Vec(2*x^4 / ((1 - x)^3*(1 + x)^2*(1 + x^2)^2) + O(x^40))) \\ Colin Barker, Nov 26 2017
(Magma) [Floor(n/4)*Ceiling((n+1)/4) : n in [0..100]]; // Wesley Ivan Hurt, Nov 26 2017
(Python)
def A038714(n): return (m:=n>>2)*(m+1) # Chai Wah Wu, Jan 18 2023

Crossrefs

Cf. A002378, A004526.

Sequence in context: A048764 A327663 A248782 * A139554 A366746 A376947

Adjacent sequences: A038711 A038712 A038713 * A038715 A038716 A038717

Keyword

nonn,easy

Author

N. J. A. Sloane, May 02 2000

Extensions

Typo in definition fixed by Harvey P. Dale, Jun 12 2022

Status

approved