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URL: https://oeis.org/A034008

⇱ A034008 - OEIS

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A034008
a(n) = floor(2^|n-1|/2). Or: 1, 0, followed by powers of 2.
36
1, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648
OFFSET
0,4
COMMENTS
Powers of 2 with additional first two terms.
Essentially the same as A131577 (and A000079).
[(-1)^n*a(n)] = [1, 0, 1, -2, 4, -8, 16, -32, ...] is the inverse binomial transform of A008619 = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...]. - Philippe Deléham, Nov 15 2009
Number of compositions (ordered partitions) of n into an even number of parts. - Geoffrey Critzer, Mar 28 2010
Number of compositions of n into an even number of even parts.
Number of compositions of n into parts k >= 2 where there are k - 1 sorts of part k. - Joerg Arndt, Sep 30 2012
Taking n-th differences of this sequence reproduces the same sequence except for a(1) = n mod 2 (parity of n) and a(0) = (-1)^a(1)*floor(n/2 + 1). - M. F. Hasler, Jan 13 2015
REFERENCES
Richard P. Stanley, Enumerative Combinatorics, Vol. I, Cambridge University Press, 1997, p. 45, exercise 9.
FORMULA
a(n) = 2^(n-2), n >= 2; a(0) = 1, a(1) = 0.
G.f.: (1-x)^2/(1-2*x).
G.f.: 1/( 1 - Sum_{k >= 1} (k-1)*x^k ). - Joerg Arndt, Sep 30 2012
G.f.: x*G(0), where G(k) = 1 + 1/(1 - (1 - x)/(1 + x*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 01 2013
a(n+1) = A131577(n) and a(n+2) = A000079(n) for all n >= 0. - M. F. Hasler, Jan 13 2015
Inverse binomial transform of (3^n - 2*n + 1)/2 for n >= 0. - Paul Curtz, Sep 24 2019
E.g.f.: (1/4)*(3 + exp(2*x) - 2*x). - Stefano Spezia, Sep 25 2019
Binomial transform of A001057(n+1) or (-1)^n*A008619(n). - Paul Curtz, Oct 07 2019
MAPLE
A034008:=n->2^(n-2): 1, 0, seq(A034008(n), n=2..50); # Wesley Ivan Hurt, Apr 12 2017
MATHEMATICA
a = x/(1 - x); CoefficientList[Series[1/(1 - a^2), {x, 0, 30}], x] (* Geoffrey Critzer, Mar 28 2010 *)
PROG
(PARI) a(n)=if(n<2, n==0, 2^(n-2))
CROSSREFS
KEYWORD
easy,nonn
EXTENSIONS
Additional comments from Barry E. Williams, May 27 2000
Additional comments from Michael Somos, Jun 18 2002
Edited by M. F. Hasler, Jan 13 2015
STATUS
approved