A008497 a(n) = floor(n/5)*floor((n+1)/5).

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 4, 4, 4, 4, 6, 9, 9, 9, 9, 12, 16, 16, 16, 16, 20, 25, 25, 25, 25, 30, 36, 36, 36, 36, 42, 49, 49, 49, 49, 56, 64, 64, 64, 64, 72, 81, 81, 81, 81, 90, 100, 100, 100, 100, 110, 121, 121, 121, 121

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A002266.

Programs

GAP

List([0..55], n-> Int(n/5)*Int((n+1)/5) ); # G. C. Greubel, Nov 08 2019

Magma

[&*[Floor((n+j)/5): j in [0..1]]: n in [0..55]]; // G. C. Greubel, Nov 08 2019

Maple

seq( mul(floor((n+j)/5), j=0..1), n=0..55); # G. C. Greubel, Nov 08 2019

Mathematica

Times@@@Partition[Floor[Range[0,60]/5],2,1] (* or *) LinearRecurrence[ {1,0,0,0,2,-2,0,0,0,-1,1},{0,0,0,0,0,1,1,1,1,2,4},60] (* Harvey P. Dale, Feb 01 2015 *)
Product[Floor[(Range[55] +j-1)/5], {j,0,1}] (* G. C. Greubel, Nov 08 2019 *)

PARI

a(n) = (n\5)*((n+1)\5); \\ Michel Marcus, Jan 06 2017

PARI

vector(56, n, prod(j=0,1, (n+j-1)\5) ) \\ G. C. Greubel, Nov 08 2019

Sage

[product(floor((n+j)/5) for j in (0..1)) for n in (0..55)] # G. C. Greubel, Nov 08 2019

Formula

From R. J. Mathar, Apr 16 2010: (Start)

a(n) = A002266(n)*A002266(n+1).

a(n)= a(n-1) + 2*a(n-5) - 2*a(n-6) - a(n-10) + a(n-11).

G.f.: x^5*(1+x^4)/ ((x^4+x^3+x^2+x+1)^2 * (1-x)^3). (End)

From Amiram Eldar, May 10 2025: (Start)

Sum_{n>=5} 1/a(n) = 2*Pi^2/3 + 1.

Sum_{n>=5} (-1)^(n+1)/a(n) = 2*log(2)-1. (End)