A068217 Denominators of coefficients in 3*log(sqrt(1+x)) power series.

2, 4, 2, 8, 10, 4, 14, 16, 6, 20, 22, 8, 26, 28, 10, 32, 34, 12, 38, 40, 14, 44, 46, 16, 50, 52, 18, 56, 58, 20, 62, 64, 22, 68, 70, 24, 74, 76, 26, 80, 82, 28, 86, 88, 30, 92, 94, 32, 98, 100, 34, 104, 106, 36, 110, 112, 38, 116, 118, 40, 122, 124, 42, 128, 130, 44, 134

Offset

1,1

Comments

Numerators are b(n)=(-1)^(n+1) if n==0 (mod 3) b(n)=(-1)^(n+1)*3 otherwise.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Formula

a(n) = 2*n/3 if n==0 (mod 3), a(n) = 2*n, otherwise.

3*log(sqrt(1+x)) = (3/2)*log(1+x) = -3 * Sum_{k>=1} (-x)^k/(2*k).

a(n) = 2*A051176(n). - Mitch Harris, Jun 29 2005 [corrected by Kevin Ryde, Oct 30 2021]

From Elmo R. Oliveira, May 08 2026: (Start)

a(n) = 2*a(n-3) - a(n-6) for n > 6.

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

Mathematica

a[n_] := If[Mod[n, 3] == 0, 2*n/3, 2*n]; Array[a, 100]  (* G. C. Greubel, Sep 21 2018 *)

Prog

(PARI) for(n=1, 100, print1(if(Mod(n, 3)==0, 2*n/3, 2*n), ", ")) \\ G. C. Greubel, Sep 21 2018

Crossrefs

Cf. A051176 (half).

Sequence in context: A296429 A396513 A065286 * A303603 A308044 A319252

Adjacent sequences: A068214 A068215 A068216 * A068218 A068219 A068220

Keyword

easy,frac,nonn

Author

Benoit Cloitre, Mar 30 2002

Extensions

Log expression corrected by Kevin Ryde, Nov 01 2021

Status

approved