A036486 a(n) = ceiling((n^3)/2).

0, 1, 4, 14, 32, 63, 108, 172, 256, 365, 500, 666, 864, 1099, 1372, 1688, 2048, 2457, 2916, 3430, 4000, 4631, 5324, 6084, 6912, 7813, 8788, 9842, 10976, 12195, 13500, 14896, 16384, 17969, 19652, 21438, 23328, 25327, 27436, 29660, 32000, 34461, 37044

Offset: 0

Author

N. J. A. Sloane, Dec 11 1999

Keywords

• nonn
• easy

Comments

a(n) is the number of compositions of even natural numbers into 3 parts < n. For example, a(2)=4 because compositions of even natural numbers into 3 parts < 2 are (0,0,0), (0,1,1), (1,0,1), and (1,1,0). a(3)=14 because compositions of even natural numbers into 3 parts <= 3 - 1 = 2 are (0,0,0), (0,1,1), (1,0,1), (1,1,0), (0,0,2), (0,2,0), (2,0,0), (1,1,2),(1,2,1),(2,1,1),(0,2,2),(2,0,2),(2,2,0) and (2,2,2). - Adi Dani, Jun 05 2011

Also the number of balls in a body-centered lattice cube with n layers. - K. G. Stier, Dec 26 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Cf. A036487.

Programs

Magma

[(2*n^3-(-1)^n+1)/4: n in [0..40]]; // Vincenzo Librandi, Jun 07 2011

Maple

[ seq(ceil((n^3)/2), n=0..100) ];
with (combinat):seq(count(Partition((n^3+1)), size=2), n=0..40); # Zerinvary Lajos, Mar 28 2008

Mathematica

Table[Ceiling[n^3/2], {n, 0, 40}] (* Wesley Ivan Hurt, May 21 2014 *)
LinearRecurrence[{3,-2,-2,3,-1},{0,1,4,14,32},50] (* Harvey P. Dale, Jan 14 2019 *)

PARI

a(n)=(2*n^3-(-1)^n+1)/4 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: x*(1+x+4*x^2) / ( (1+x)*(x-1)^4 ). - R. J. Mathar, Jun 06 2011

a(n) = (2*n^3 - (-1)^n + 1)/4. - Bruno Berselli, Jun 07 2011

a(n) = n^3 - A036487(n), where n^3 is the number of compositions of natural numbers into 3 parts < n. - R. J. Mathar, Jun 07 2011

a(n) = (n^3 + (n mod 2))/2. - Wesley Ivan Hurt, May 21 2014

E.g.f.: (x*(1 + 3*x + x^2)*cosh(x) + (1 + x + 3*x^2 + x^3)*sinh(x))/2. - Stefano Spezia, Sep 09 2022