A160791 a(n) = binomial(N, n - N) where N = 1 + floor(n/2).

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, 14, 105, 15, 120, 16, 136, 17, 153, 18, 171, 19, 190, 20, 210, 21, 231, 22, 253, 23, 276, 24, 300, 25, 325, 26, 351, 27, 378, 28, 406, 29, 435, 30, 465

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Formula

From R. J. Mathar, Feb 09 2010: (Start)

a(2n+1) = n+1 and a(2n) = A000217(n) with a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).

G.f.: x*(1+x-x^2)/(1-x^2)^3. (End)

a(n) = (n^2+6*n+4+(n^2-2*n-4)*(-1)^n)/16. - Luce ETIENNE, Mar 31 2015

E.g.f.: (x*(x+4)*cosh(x) + (3*x+4)*sinh(x))/8. - G. C. Greubel, Apr 26 2018

Maple

a := proc(n) 1 + floor(n/2); binomial(%, n - %) end:
seq(a(n), n = 0..60);  # Peter Luschny, Jul 02 2024

Mathematica

Join[{0}, Riffle[Range[30], Range[30] (Range[30] + 1)/2]] (* Bruno Berselli, Jul 15 2013 *)
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 1, 1, 2, 3, 3, 6}, 60] (* Vincenzo Librandi, Apr 02 2015 *)

Prog

(PARI) Vec(x*(1+x-x^2)/(1-x^2)^3 + O(x^80)) \\ Michel Marcus, Apr 01 2015
(Magma) [(n^2+6*n+4+(n^2-2*n-4)*(-1)^n)/16: n in [0..70]]; // Vincenzo Librandi, Apr 02 2015

Crossrefs

First differences of A160790.

Cf. A160791, A160792, A160793.

Sequence in context: A372808 A266286 A045892 * A115973 A057047 A361168

Adjacent sequences: A160788 A160789 A160790 * A160792 A160793 A160794

Keyword

easy,nonn

Author

Omar E. Pol, May 29 2009

Extensions

a(0) = 0 prepended and new name by Peter Luschny, Jul 02 2024

Status

approved