%I #39 Jul 02 2024 08:21:57
%S 0,1,1,2,3,3,6,4,10,5,15,6,21,7,28,8,36,9,45,10,55,11,66,12,78,13,91,
%T 14,105,15,120,16,136,17,153,18,171,19,190,20,210,21,231,22,253,23,
%U 276,24,300,25,325,26,351,27,378,28,406,29,435,30,465
%N a(n) = binomial(N, n - N) where N = 1 + floor(n/2).
%H G. C. Greubel, <a href="/A160791/b160791.txt">Table of n, a(n) for n = 0..5000</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 From _R. J. Mathar_, Feb 09 2010: (Start)
%F a(2n+1) = n+1 and a(2n) = A000217(n) with a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
%F G.f.: x*(1+x-x^2)/(1-x^2)^3. (End)
%F a(n) = (n^2+6*n+4+(n^2-2*n-4)*(-1)^n)/16. - _Luce ETIENNE_, Mar 31 2015
%F E.g.f.: (x*(x+4)*cosh(x) + (3*x+4)*sinh(x))/8. - _G. C. Greubel_, Apr 26 2018
%p a := proc(n) 1 + floor(n/2); binomial(%, n - %) end:
%p seq(a(n), n = 0..60); # _Peter Luschny_, Jul 02 2024
%t Join[{0}, Riffle[Range[30], Range[30] (Range[30] + 1)/2]] (* _Bruno Berselli_, Jul 15 2013 *)
%t LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 1, 1, 2, 3, 3, 6}, 60] (* _Vincenzo Librandi_, Apr 02 2015 *)
%o (PARI) Vec(x*(1+x-x^2)/(1-x^2)^3 + O(x^80)) \\ _Michel Marcus_, Apr 01 2015
%o (Magma) [(n^2+6*n+4+(n^2-2*n-4)*(-1)^n)/16: n in [0..70]]; // _Vincenzo Librandi_, Apr 02 2015
%Y First differences of A160790.
%Y Cf. A160791, A160792, A160793.
%K easy,nonn
%O 0,4
%A _Omar E. Pol_, May 29 2009
%E a(0) = 0 prepended and new name by _Peter Luschny_, Jul 02 2024