A010762 a(n) = floor(n/2) * floor(n/3).

0, 0, 1, 2, 2, 6, 6, 8, 12, 15, 15, 24, 24, 28, 35, 40, 40, 54, 54, 60, 70, 77, 77, 96, 96, 104, 117, 126, 126, 150, 150, 160, 176, 187, 187, 216, 216, 228, 247, 260, 260, 294, 294, 308, 330, 345, 345, 384, 384, 400, 425, 442, 442, 486, 486, 504, 532, 551

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

Comments

a(n) is also the number of 5 boxes polyomino (invert U patterns) packing into n X n square. The 6 boxes 2 X 3 (rectangular patterns) also gives the same sequence but difference in squares left. See illustration in links. - Kival Ngaokrajang, Nov 10 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Kival Ngaokrajang, Illustration of initial terms of invert u and 2 X 3 rectangular patterns.
• Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1,1,0,-1,-1,0,1).

Crossrefs

Cf. A002264, A004526, A010761, A110533, A242669.

Programs

Magma

[Floor(n/2)*Floor(n/3) : n in [1..50]]; // Wesley Ivan Hurt, Jun 22 2014

Maple

[ seq(floor(n/2)*floor(n/3), n=1..64) ];

Mathematica

Table[Floor[n/2]*Floor[n/3], {n, 1, 70}] (* Clark Kimberling, May 18 2012 *)
CoefficientList[Series[- x^2 (x^7 + x^6 + x^5 + 2 x^4 + 3 x^3 + x^2 + 2 x+1)/((x - 1)^3 (x + 1)^2 (x^2 - x + 1) (x^2 + x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2013 *)
LinearRecurrence[{0,1,1,0,-1,1,0,-1,-1,0,1},{0,0,1,2,2,6,6,8,12,15,15},60] (* Harvey P. Dale, Jan 09 2016 *)

PARI

a(n) = (n\2) * (n\3) \\ Charles R Greathouse IV, Oct 07 2015; corrected by Michel Marcus, Jun 01 2025

Formula

a(n) = A004526(n) * A002264(n). - Reinhard Zumkeller, Jul 25 2005

a(n) = a(n-2) + a(n-3) - a(n-5) + a(n-6) - a(n-8) - a(n-9) + a(n-11). - Clark Kimberling, May 18 2012

G.f.: -x^3*(x^7+x^6+x^5+2*x^4+3*x^3+x^2+2*x+1) / ((x-1)^3*(x+1)^2*(x^2-x+1)*(x^2+x+1)^2). - Colin Barker, Apr 05 2013

Sum_{n>=3} (-1)^(n+1)/a(n) = sqrt(3)*Pi/4 + 9*log(3)/4 - 2*log(2) - 3/2. - Amiram Eldar, Mar 30 2023

E.g.f.: (3*(x - 1)*x*cosh(x) - sqrt(3)*exp(-x/2)*(1 + exp(x) + 4*x)*sin(sqrt(3)*x/2)/2 + 3*cos(sqrt(3)*x/2)*sinh(x/2) + 3*(1 + x^2)*sinh(x))/18. - Stefano Spezia, Jun 01 2025