

A165998


Denominators of Taylor series expansion of 1/(3*x)*log((1+x)/(1x)^2)


27



1, 6, 3, 12, 5, 18, 7, 24, 9, 30, 11, 36, 13, 42, 15, 48, 17, 54, 19, 60, 21, 66, 23, 72, 25, 78, 27, 84, 29, 90, 31, 96, 33, 102, 35, 108, 37, 114, 39, 120, 41, 126, 43, 132, 45, 138, 47, 144, 49, 150, 51, 156, 53, 162, 55, 168, 57, 174, 59, 180, 61, 186, 63, 192, 65, 198
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OFFSET

0,2


COMMENTS

Numerators are all 1.
Setting x=1/3 into
1/(3*x)*log((1+x)/(1x)^2) = Sum_{k>=0} x^k/((2(1)^k)*(k+1)),
log(3) = Sum_{k>=0} 1/((2(1)^k)*(k+1)*3^k)
= Sum_{k>=0} (9/(2k+1)+1/(2k+2))/9^(k+1)
is obtained.
It appears that this is also the first differences of the generalized decagonal numbers A074377.  Omar E. Pol, Sep 10 2011
It appears that this is also A005408 and positive terms of A008588 interleaved.  Omar E. Pol, May 28 2012
With offset 1 this sequence is multiplicative (in fact, a generalized totient function): a(p^e) = p^e for any odd prime p and a(2^e) = 3*2^e for e >= 1.  Charles R Greathouse IV, Mar 09 2015


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1004
TianXiao He, Peter J.S. Shiue, Zihan Nie, Minghao Chen, Recursive sequences and GirardWaring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 10491062.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1)


FORMULA

G.f.: (1+6*x+x^2)/(1x^2)^2.
a(n) = (2(1)^n)*(n+1) (see PARI's code by Jaume Oliver Lafont).
a(2n)= 2n+1. a(2n+1) = 6*(n+1).  R. J. Mathar, Apr 02 2011


MATHEMATICA

LinearRecurrence[{0, 2, 0, 1}, {1, 6, 3, 12}, 50] (* Vincenzo Librandi, Feb 22 2012 *)


PROG

(PARI) a(n)=(2(1)^n)*(n+1)
(Magma) [(2(1)^n)*(n+1): n in [0..350]]; // Vincenzo Librandi, Apr 04 2011


CROSSREFS

Cf. A154920.
Sequence in context: A131894 A335393 A040033 * A322091 A329583 A050132
Adjacent sequences: A165995 A165996 A165997 * A165999 A166000 A166001


KEYWORD

frac,nonn,easy


AUTHOR

Jaume Oliver Lafont, Oct 03 2009


STATUS

approved



