A008217 a(n) = floor(n/4)*floor((n+1)/4).

0, 0, 0, 0, 1, 1, 1, 2, 4, 4, 4, 6, 9, 9, 9, 12, 16, 16, 16, 20, 25, 25, 25, 30, 36, 36, 36, 42, 49, 49, 49, 56, 64, 64, 64, 72, 81, 81, 81, 90, 100, 100, 100, 110, 121, 121, 121, 132, 144, 144, 144, 156, 169, 169, 169, 182, 196, 196, 196, 210, 225, 225, 225, 240, 256, 256, 256, 272, 289, 289, 289, 306

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Oblong numbers, squares and quarter-squares are subsequences: a(A004767(n)) = A002378(n); a(A008586(n)) = A000290(n); a(A005408(n)) = A002620(n). - Reinhard Zumkeller, Oct 09 2011

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,0,-2,2,-2,1).

Crossrefs

Cf. A000290, A002378, A002620, A004767, A005408, A008133, A008586.

Programs

Haskell

a008217 n = a008217_list !! n
a008217_list = zipWith (*) (tail qs) qs where qs = map (`div` 4) [0..]
-- Reinhard Zumkeller, Oct 09 2011

Mathematica

a[n_] := Floor[n/4] * Floor[(n+1)/4]; Array[a, 100, 0] (* Amiram Eldar, May 10 2025 *)
LinearRecurrence[{2,-2,2,0,-2,2,-2,1},{0,0,0,0,1,1,1,2},80] (* Harvey P. Dale, Aug 18 2025 *)

PARI

a(n) = floor(n/4)*floor((n+1)/4); /* Joerg Arndt, Mar 31 2013 */

Python

def A008217(n): return (n>>2)*(n+1>>2) # Chai Wah Wu, Feb 02 2023

Formula

G.f.: -x^4*(x^2-x+1) / ((x-1)^3*(x+1)*(x^2+1)^2). - Colin Barker, Mar 31 2013

From Amiram Eldar, May 10 2025: (Start)

Sum_{n>=4} 1/a(n) = Pi^2/2 + 1.

Sum_{n>=4} (-1)^n/a(n) = Pi^2/6 - 1. (End)