A045896 Denominator of n/((n+1)*(n+2)) = A026741/A045896.

1, 6, 6, 20, 15, 42, 28, 72, 45, 110, 66, 156, 91, 210, 120, 272, 153, 342, 190, 420, 231, 506, 276, 600, 325, 702, 378, 812, 435, 930, 496, 1056, 561, 1190, 630, 1332, 703, 1482, 780, 1640, 861, 1806, 946, 1980, 1035, 2162, 1128, 2352, 1225, 2550, 1326, 2756, 1431

Offset: 0

Author

Ralf W. Grosse-Kunstleve, Dec 11 1999

Keywords

• nonn
• easy
• frac
• nice

Comments

Also period length divided by 2 of pairs (a,b), where a has period 2*n-2 and b has period n.

From Paul Curtz, Apr 17 2014: (Start)

Difference table of A026741/A045896:

0, 1/6, 1/6, 3/20, 2/15, 5/42, ...

1/6, 0, -1/60, -1/60, -1/70, -1/84, ... = 1/6, -A051712/A051713

-1/6, -1/60, 0, 1/420, 1/420, 1/504, ...

3/20, 1/60, 1/420, 0, -1/2520, -1/2520, ...

-2/15, -1/70, -1/420, -1/2520, 0, 1/13860, ...

5/42, 1/84, 1/504, 1/2520, -1/13860, 0, ...

Autosequence of the first kind. The main diagonal is A000004. The first two upper diagonals are equal. Their denominators are A000911. (End)

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Ralf W. Grosse-Kunstleve, Origin of EIS sequences A045895 & A045896. [Wayback Machine copy]
• Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), Article 00.2.9.

Crossrefs

Factor of A160466.

Cf. A000004, A000384, A000911, A045895, A026741, A051714, A241269.

Programs

Haskell

import Data.Ratio ((%), denominator)
a045896 n = denominator $ n % ((n + 1) * (n + 2))
-- Reinhard Zumkeller, Dec 12 2011

Maple

seq((n+1)*(n+2)*(3-(-1)^n)/4, n=0..20); # C. Ronaldo
with(combinat): seq(lcm(n+1,binomial(n+2,n)), n=0..50); # Zerinvary Lajos, Apr 20 2008

Mathematica

Table[LCM[2*n + 2, n + 2]/2, {n, 0, 40}] (* corrected by Amiram Eldar, Sep 14 2022 *)
Denominator[#[[1]]/(#[[2]]#[[3]])&/@Partition[Range[0,60],3,1]] (* Harvey P. Dale, Aug 15 2013 *)

PARI

Vec((2*x^3+3*x^2+6*x+1)/(1-x^2)^3+O(x^99)) \\ Charles R Greathouse IV, Mar 23 2016

Formula

G.f.: (2*x^3+3*x^2+6*x+1)/(1-x^2)^3.

a(n) = (n+1)*(n+2) if n odd; or (n+1)*(n+2)/2 if n even = (n+1)*(n+2)*(3-(-1)^n)/4. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004

a(2*n) = A000384(n+1); a(2*n+1) = A026741(n+1). - Reinhard Zumkeller, Dec 12 2011

Sum_{n>=0} 1/a(n) = 1 + log(2). - Amiram Eldar, Sep 11 2022

From Amiram Eldar, Sep 14 2022: (Start)

a(n) = lcm(2*n+2, n+2)/2.

a(n) = A045895(n+2)/2. (End)

E.g.f.: (2 + 8*x + x^2)*cosh(x)/2 + (2 + 2*x + x^2)*sinh(x). - Stefano Spezia, Apr 24 2024