A367965 - OEIS

%I #15 Dec 06 2024 07:01:02

%S 0,1,5,4,16,9,33,16,56,25,85,36,120,49,161,64,208,81,261,100,320,121,

%T 385,144,456,169,533,196,616,225,705,256,800,289,901,324,1008,361,

%U 1121,400,1240,441,1365,484,1496,529,1633,576,1776,625,1925,676,2080,729,2241,784

%N a(n) = (1/8)*(4*n^2 + 6*n + (-1)^n*(2*n*(n + 1) - 1) + 1).

%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) = Sum_{k=0..n} (-1)^(n - k) * A367964(n, k).

%F a(2*n) = n*(3*n+2) = A045944(n).

%F a(2*n-1) = n^2 = A000290(n).

%F G.f.: x*(1 + 5*x + x^2 + x^3)/(1 - x)^3*(1 + x)^3). - _Stefano Spezia_, Dec 07 2023

%F Sum_{n>=1} 1/a(n) = Pi^2/6 + Pi/(4*sqrt(3)) - 3*(log(3)-1)/4. - _Amiram Eldar_, Dec 06 2024

%p a := n -> (1/8)*(4*n^2 + 6*n + (-1)^n*(2*n*(n + 1) - 1) + 1):

%p seq(a(n), n = 0..55);

%t LinearRecurrence[{0,3,0,-3,0,1},{0,1,5,4,16,9},100] (* _Paolo Xausa_, Dec 07 2023 *)

%Y Cf. A367964, A000290, A045944.

%K nonn,easy

%O 0,3

%A _Peter Luschny_, Dec 07 2023