%I #36 Jan 19 2023 02:14:36
%S 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,
%T 42,56,56,56,56,72,72,72,72,90,90,90,90,110,110,110,110,132,132,132,
%U 132,156,156,156,156,182,182,182,182,210,210,210,210,240
%N Pronic numbers repeated 4 times; a(n) = floor(n/4) * ceiling((n+1)/4).
%C From _Wesley Ivan Hurt_, Nov 25 2017: (Start)
%C 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.
%C 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)
%H Colin Barker, <a href="/A038714/b038714.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1).
%F 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
%F From _Wesley Ivan Hurt_, Nov 25 2017: (Start)
%F a(n) = floor(n/4) * (floor(n/4) + 1).
%F a(n) = Sum_{i=1..floor(n/2)} i * ((i+1) mod 2).
%F (End)
%F G.f.: 2*x^4 / ((1 - x)^3*(1 + x)^2*(1 + x^2)^2). - _Colin Barker_, Nov 26 2017
%F a(n) = A002378(A004526(n)). - _Wesley Ivan Hurt_, Nov 26 2017
%F 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
%p A038714:=n->floor(n/4)*ceil((n+1)/4): seq(A038714(n), n=0..100); # _Wesley Ivan Hurt_, Nov 26 2017
%t Table[Floor[n/4] Ceiling[(n + 1)/4], {n, 0, 100}] (* _Wesley Ivan Hurt_, Nov 26 2017 *)
%t 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 *)
%o (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
%o (Magma) [Floor(n/4)*Ceiling((n+1)/4) : n in [0..100]]; // _Wesley Ivan Hurt_, Nov 26 2017
%o (Python)
%o def A038714(n): return (m:=n>>2)*(m+1) # _Chai Wah Wu_, Jan 18 2023
%Y Cf. A002378, A004526.
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_, May 02 2000
%E Typo in definition fixed by _Harvey P. Dale_, Jun 12 2022