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) = (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
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
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