OFFSET
0,4
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
J. Heading, Theorem relating to the development of a reflection coefficient in terms of a small parameter, J. Phys. A 14 (1981), 357-367.
Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text). - N. J. A. Sloane, Mar 26 2015
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).
FORMULA
From Paul Barry, Apr 28 2004: (Start)
G.f.: x*(1-x)/((1-2*x)*(1-2*x^2)).
a(n) = 2*a(n-1)+2*a(n-2)-4*a(n-3).
a(n) = 2^n/2-2^(n/2)*(1+(-1)^n)/4. (End)
G.f.: (1+x*Q(0))*x/(1-x), where Q(k)= 1 - 1/(2^k - 2*x*2^(2*k)/(2*x*2^k - 1/(1 + 1/(2*2^k - 8*x*2^(2*k)/(4*x*2^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013
EXAMPLE
From Gus Wiseman, Feb 26 2022: (Start)
Also the number of integer compositions of n with at least one odd part. For example, the a(1) = 1 through a(5) = 16 compositions are:
(1) (1,1) (3) (1,3) (5)
(1,2) (3,1) (1,4)
(2,1) (1,1,2) (2,3)
(1,1,1) (1,2,1) (3,2)
(2,1,1) (4,1)
(1,1,1,1) (1,1,3)
(1,2,2)
(1,3,1)
(2,1,2)
(2,2,1)
(3,1,1)
(1,1,1,2)
(1,1,2,1)
(1,2,1,1)
(2,1,1,1)
(1,1,1,1,1)
(End)
MAPLE
f := n-> if n mod 2 = 0 then 2^(n-1)-2^((n-2)/2) else 2^(n-1); fi;
MATHEMATICA
LinearRecurrence[{2, 2, -4}, {0, 1, 1}, 30] (* Harvey P. Dale, Nov 30 2015 *)
Table[2^(n-1)-If[EvenQ[n], 2^(n/2-1), 0], {n, 0, 15}] (* Gus Wiseman, Feb 26 2022 *)
PROG
(Magma) [Floor(2^n/2-2^(n/2)*(1+(-1)^n)/4): n in [0..40]]; // Vincenzo Librandi, Aug 20 2011
(PARI) Vec(x*(1-x)/((1-2*x)*(1-2*x^2)) + O(x^50)) \\ Michel Marcus, Jan 28 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved