A237109 a(n) is the numerator of 2*n / ((n+2) * (n+3)).

1, 1, 1, 4, 5, 1, 7, 8, 3, 5, 11, 4, 13, 7, 5, 16, 17, 3, 19, 20, 7, 11, 23, 8, 25, 13, 9, 28, 29, 5, 31, 32, 11, 17, 35, 12, 37, 19, 13, 40, 41, 7, 43, 44, 15, 23, 47, 16, 49, 25, 17, 52, 53, 9, 55, 56, 19, 29, 59, 20, 61, 31, 21, 64, 65, 11, 67, 68, 23, 35, 71, 24

Offset: 1

Author

Paul Curtz, Feb 03 2014

Keywords

• nonn
• frac
• mult

Comments

Previous name was: Numerators of the third row of the Akiyama-Tanigawa algorithm (or transformation) applied to A001008(n+1)/A002805(n+1).

Successive rows:

3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, ...;

-1/3, -1/2, -3/5, -2/3, -5/7, -3/4, -7/9, -4/5, ... = A026741(n+1)/A026741(n+3);

1/6, 1/5, 1/5, 4/21, 5/28, 1/6, 7/45, 8/55, 3/22, ...;

-1/30, 0, ...;

-1/30.

First column denominators: 2,3,6,30,30,... = A051717(n+1).

A001008(n)/A002805(n) is the inverse Akiyama-Tanigawa transformation applied to A027641(n)/A027642(n). A051716(n)/A051717(n) comes from 0 followed by A164555(n)/A027642(n). Then, from the two Bernoulli numbers.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..2500

Programs

Magma

[Numerator(2*n/((n+2)*(n+3))): n in [1..50]]; // G. C. Greubel, Aug 07 2018

Mathematica

a[1, n_] := HarmonicNumber[n+1]; a[n_, m_] := a[n, m] = m*(a[n-1, m]-a[n-1, m+1]); Table[a[3, m] // Numerator, {m, 1, 72}] (* Jean-François Alcover, Feb 11 2014 *)
a[ n_] := n / {1, 2, 3, 1, 1, 6, 1, 1, 3, 2, 1, 3}[[Mod[n, 12, 1]]]; (* Michael Somos, Aug 01 2017 *)

PARI

{a(n) = if( n<0, -a(-n), numerator( 2*n / ((n+2) * (n+3))))}; /* Michael Somos, Aug 01 2017 */

Formula

a(n) = -a(-n) for all n in Z. - Michael Somos, Aug 01 2017

From Amiram Eldar, Nov 17 2022: (Start)

Multiplicative with a(2) = 1, a(2^e) = 2^e for e > 1, a(3^e) = 3^(e-1), and a(p^e) = p^e for p >= 5.

Sum_{k=1..n} a(k) ~ (49/144) * n^2. (End)

Dirichlet g.f.: zeta(s-1)*(1-1/2^s+2/4^s)*(1-2/3^s). - Amiram Eldar, Jan 05 2023

Extensions

New name using Somos's Pari code from Joerg Arndt, May 27 2018

Keyword:mult added by Andrew Howroyd, Jul 31 2018