A359366 - OEIS

%I #21 Apr 28 2025 19:10:17

%S 0,1,3,4,10,9,21,16,36,25,55,36,78,49,105,64,136,81,171,100,210,121,

%T 253,144,300,169,351,196,406,225,465,256,528,289,595,324,666,361,741,

%U 400,820,441,903,484,990,529,1081,576,1176,625,1275,676,1378,729,1485

%N a(n) = (1/8)*(((3*n + 1) + (n - 1)*(-1)^n)*(n + 1)).

%H Daniel Mondot, <a href="/A359366/b359366.txt">Table of n, a(n) for n = 0..9999</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).

%F a(n) = ((n + 1) / 2)^2 if n is odd, otherwise (n^2 + n) / 2.

%F a(n) = [x^n] -(x*(x^3 + x^2 + 3*x + 1))/(x^2 - 1)^3.

%F a(n) = n! * [x^n] (1/4)*((1 + x*(x + 4))*sinh(x) + x*(2*x + 3)*cosh(x)).

%p a := n -> (1/8)*(((3*n + 1) + (n - 1)*(-1)^n)*(n + 1)):

%p # Or:

%p a := n -> ifelse(irem(n, 2) = 1, ((n + 1) / 2)^2, (n^2 + n)/2):

%p seq(a(n), n = 0..54);

%t a[n_] := (1/8)*(((3*n + 1) + (n - 1)*(-1)^n)*(n + 1)); Array[a,55,0] (* _Stefano Spezia_, Apr 28 2025 *)

%Y Cf. A000290, A014105, A106465, A056136.

%K nonn,easy

%O 0,3

%A _Peter Luschny_, Dec 30 2022