|
|
A038714
|
|
Pronic numbers repeated 4 times; a(n) = floor(n/4) * ceiling((n+1)/4).
|
|
1
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
COMMENTS
|
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
|
|
|
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
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)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|