A061038 Denominator of 1/4 - 1/n^2.

1, 36, 16, 100, 9, 196, 64, 324, 25, 484, 144, 676, 49, 900, 256, 1156, 81, 1444, 400, 1764, 121, 2116, 576, 2500, 169, 2916, 784, 3364, 225, 3844, 1024, 4356, 289, 4900, 1296, 5476, 361, 6084, 1600, 6724, 441, 7396, 1936, 8100, 529, 8836

Offset: 2

Author

N. J. A. Sloane, May 26 2001

Keywords

• nonn
• frac
• nice
• easy

Links

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

Crossrefs

See A061037 for comments, references, links.

Cf. A145979. - Bruno Berselli, Mar 21 2011

Programs

Haskell

import Data.Ratio ((%), denominator)
a061038 n = denominator (1%4 - 1%n^2)  -- Reinhard Zumkeller, Jan 22 2012

Magma

[ Denominator(1/4-1/n^2): n in [2..50] ]; // Vincenzo Librandi, Feb 10 2011

Mathematica

Table[Denominator[1/4 - 1/n^2], {n, 2, 60}] (* Stefan Steinerberger, Apr 08 2006 *)
LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{1,36,16,100,9,196,64,324, 25,484,144,676},50] (* Harvey P. Dale, Aug 05 2018 *)

PARI

a(n) = { denominator(1/4 - 1/n^2) } \\ Harry J. Smith, Jul 17 2009

SageMath

def A061038(n): return denominator(1/4 - 1/n^2)
[A061038(n) for n in range(2,51)] # G. C. Greubel, Apr 18 2023

Formula

a(4n+2) = (2n+1)^2, a(2n+3) = (4n+6)^2, a(4n+4) = (4n+4)^2. - Ralf Stephan, Jun 10 2005

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12). - Paul Curtz, Feb 25 2011

From Bruno Berselli, Mar 21 2011: (Start)

G.f.: x^2*(1 +36*x +16*x^2 +100*x^3 +6*x^4 +88*x^5 +16*x^6 +24*x^7 +x^8 +4*x^9 +4*x^11)/(1-x^4)^3.

a(n) = (1/64)*( n*(16 - (1+(-1)^n)*(5-i^n)) )^2 with i=sqrt(-1).

a(n) = (n/(n-4))^2 * a(n-4) for n>5. (End)

a(n) = 4*n^2 / gcd(4*n^2, (n^2-4)). - Colin Barker, Jan 13 2014

Sum_{n>=2} 1/a(n) = Pi^2/6 - 1/4. - Amiram Eldar, Aug 12 2022

Extensions

More terms from Stefan Steinerberger, Apr 08 2006