OFFSET
0,2
COMMENTS
Previous name was: A product form based on the Fibonacci product form: f(n) = 2^n*Product_{k=1..floor((n-1)/2)} (1 + 3*cos(k*Pi/n)^2).
Gary W. Adamson found this article, I experimented. Based on the paper Fibonacci identity of: f(n) = Product_{k=1..floor((n-1)/2)} (1 + 4*cos(k*Pi/n)^2). I changed the 4 to a 3 and used 2^n to get rid of the rational terms. The product comes down slow in Mathematica: I tried 30 but no luck.
For n > 0, Select an odd size subset S of {1, 2, ..., n}, then select a subset of S. - Geoffrey Critzer, Mar 03 2010
It appears that if s(n) is a first order rational sequence of the form s(1) = 2, s(n) = (s(n-1) + 2)/(2*s(n-1) + 1), n > 1 then s(n) = a(n)/(a(n) + (-1)^n). - Gary Detlefs, Nov 16 2010
For n >= 1, a(n) counts closed walks of length n + 1 on the vertex of a triangle to which two loops have been added to one of remaining vertices. - David Neil McGrath, Sep 04 2014
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Dhroova Aiylam and Tanya Khovanova, Weighted Mediants and Fractals, arXiv:1711.01475 [math.NT], 2017. See p. 18.
M. H. Albert and R. Brignall, Enumerating indices of Schubert varieties defined by inclusions, arXiv:1301.3188 [math.CO], 2013. - From N. J. A. Sloane, Feb 11 2013
N. Garnier and O. Ramaré, Fibonacci numbers and trigonometric identities
N. Garnier and O. Ramaré, Fibonacci numbers and trigonometric identities, Fibonacci Quart. 46/47 (2008/2009), no. 1, 56-61.
Index entries for linear recurrences with constant coefficients, signature (2,3).
FORMULA
a(n) = 2^n * Product_{k = 1..floor((n-1)/2)} (1 + 3 * cos(k * Pi/n)^2).
From Geoffrey Critzer, Mar 03 2010: (Start)
For n > 0, a(n) = Sum_{k = 1, 3, 5, ...} C(n, k)* 2^k.
E.g.f.: 1 + sinh(2*x)*exp(x). (End)
From R. J. Mathar, Mar 11 2010: (Start)
a(n) = (3^n - (-1)^n)/2, n > 0.
G.f.: (1 - 3*x^2)/((1 + x)*(1 - 3*x)). (End)
a(n) = 2*a(n-1) + 3*a(n-2) for n >= 2, a(1) = 2, and a(2) = 4. - David Neil McGrath, Sep 04 2014
a(n) = M^n[1,2] = M^n[2,1] for n>0, where M = [1,2;2,1]. - Rigoberto Florez, May 05 2020
EXAMPLE
G.f. = 1 + 2*x + 4*x^2 + 14*x^3 + 40*x^4 + 122*x^5 + 364*x^6 + 1094*x^7 + ...
MATHEMATICA
f[n_] = 2^n Product[(1 + 3 Cos[k Pi/n]^2), {k, Floor[(n - 1)/2]}]; Table[FullSimplify[ExpandAll[f[n]]], {n, 0, 15}]
(* Second program: *)
CoefficientList[Series[(1-3x^2)/((1+x)(1-3x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 15 2014 *)
Join[{1}, LinearRecurrence[{2, 3}, {2, 4}, 30]] (* Jean-François Alcover, Jan 07 2019 *)
PROG
(Sage)
def A152011(n) :
if n == 0 : return 1
return add(2^(n-k)*binomial(n, k) for k in range(n)[::2]) # Peter Luschny, Jul 30 2012
(PARI) a(n)=floor(2^n*prod(k=1, floor((n-1)/2), 1+3*cos(k*Pi/n)^2)+1/2) \\ Edward Jiang, Sep 08 2014
(PARI) a(n)=if(n, (3^n-(-1)^n)/2, 1) \\ Charles R Greathouse IV, Sep 15 2014
(Magma) [1] cat [(3^n-(-1)^n)/2: n in [1..30]]; // Vincenzo Librandi, Sep 15 2014
CROSSREFS
KEYWORD
nonn,walk,easy
AUTHOR
Roger L. Bagula and Gary W. Adamson, Nov 19 2008
EXTENSIONS
Terms a(16)-a(25) from Peter Luschny, Jul 30 2012
New name (using R. J. Mathar's formula) by Joerg Arndt, Sep 09 2014
STATUS
approved