A003185 a(n) = (4*n+1)*(4*n+5).

5, 45, 117, 221, 357, 525, 725, 957, 1221, 1517, 1845, 2205, 2597, 3021, 3477, 3965, 4485, 5037, 5621, 6237, 6885, 7565, 8277, 9021, 9797, 10605, 11445, 12317, 13221, 14157, 15125, 16125, 17157, 18221

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Bisection of A078371. - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Nov 19 2004

a(n) is the smallest number not in the sequence such that Sum_{k=0..n} 1/a(k) has a denominator 4*n+5. - Derek Orr, Jun 21 2015

a(n) is the number of 2 X 2 matrices with all elements in {-n,..,0,..,n} with permanent = determinant^n except for a(0), where a(0)=0, but A003185(0) = 5. - Indranil Ghosh, Jan 04 2017

Links

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

Crossrefs

Cf. A001513, A003185, A063164, A078371.

Programs

Mathematica

Table[(4n+1)(4n+5),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{5,45,117},40] (* Harvey P. Dale, Jan 27 2013 *)

PARI

a(n) = (4*n+1)*(4*n+5); \\ Michel Marcus, Jan 17 2023

Python

a= lambda n: (4*n+1)*(4*n+5) # Indranil Ghosh, Jan 04 2017

Formula

1 = Sum_{n>=0} 4/a(n). Sum_{k=0..n} 4/a(k) = 4(n+1)/[4(n+1)+1]. Integral_{x=0..1} 1/(1 + x^4) = Sum_{n>=0} 4/a(2*n) = Sum_{n>=0} (-1)^n/(4n+1). - Gary W. Adamson, Jun 18 2003

1 = 1/5 + Sum_{n>=1} 16/a(n); with partial sums (4n+1)/(4n+5). - Gary W. Adamson, Jun 18 2003

From R. J. Mathar, Apr 04 2008: (Start)

O.g.f.: (-5-30*x+3*x^2)/(-1+x)^3.

a(3*n) = A001513(2*n).

Conjecture: a(n+1)-a(n) = A063164(n+2). (End)

a(n) = 32*n + a(n-1) + 8 (with a(0)=5). - Vincenzo Librandi, Nov 12 2010

a(0)=5, a(1)=45, a(2)=117, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jan 27 2013

Sum_{n>=0} (-1)^n/a(n) = (log(2*sqrt(2)+3) + Pi)/(8*sqrt(2)) - 1/4. - Amiram Eldar, Oct 08 2023