A034008 a(n) = floor(2^|n-1|/2). Or: 1, 0, followed by powers of 2.

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

Author

Wolfdieter Lang

Keywords

• easy
• nonn

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.
• Index entries for linear recurrences with constant coefficients, signature (2).

Crossrefs

Cf. A011782, A131577, A000079.

Cf. A001057, A008619.

Programs

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 *)

PARI

a(n)=if(n<2,n==0,2^(n-2))

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

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