0,3

Powers of 2 doubled up. The usual OEIS policy is to omit the duplicates in such cases (when this would become A000079). This is an exception.

Number of symmetric partitions of n: e.g. 5 = 2+1+2 = 1+3+1 = 1+1+1+1+1 so a(5) = 4; 6 = 3+3 = 2+2+2 = 1+4+1 = 2+1+1+2 = 1+2+2+1 = 1+1+2+1+1 = 1+1+1+1+1+1 so a(6) = 8. - Henry Bottomley (se16(AT)btinternet.com), Dec 10 2001

This sequence is the number of digits of each term of A061519. [From Dmitry Kamenetsky (dkamen(AT)rsise.anu.edu.au), Jan 17 2009]

Starting with offset 1 = binomial transform of [1, 1, -1, 3, -7, 17, -41,...]; where A001333 = (1, 1, 3, 7, 17, 41,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 25 2009]

Contribution from Dennis P. Walsh (dwalsh(AT)mtsu.edu), Oct 27 2009: (Start)

a(n+1) is the number of symmetric subsets of [n]={1,2,...,n}. A subset S of

[n] is symmetric if k is an element of S implies (n-k+1) is an element of S. (End)

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

E. Deutsch, Problem 1633, Math. Mag., 74 #5 (2001), p. 403.

D. Merlini, F. Uncini and M. C. Verri, A unified approach to the study of general and palindromic compositions, Integers 4 (2004), A23, 26 pp.

Index entries for sequences related to linear recurrences with constant coefficients

S. Heubach and T. Mansour, Counting rises, levels and drops in compositions

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1067

Dennis Walsh, Notes on symmetric subsets of {1, 2, ..., n} [From Dennis P. Walsh (dwalsh(AT)mtsu.edu), Oct 27 2009]

a(n) = a(n-1)*a(n-2)/a(n-3) = 2a(n-2) = 2^A004526(n). G.f.: (1+x)/(1-2x^2)

(1/2+sqrt(1/8))*sqrt(2)^n+(1/2-sqrt(1/8))*(-sqrt(2))^n. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 11 2003

E.g.f.: cosh(sqrt(2)x)+sinh(sqrt(2)x)/sqrt(2). - Paul Barry (pbarry(AT)wit.ie), Jul 16 2003

The signed sequence (-1)^n2^[n/2] has a(n)=(sqrt(2))^n(1/2-sqrt(2)/4)+(-sqrt(2))^n(1/2+sqrt(2)/4). It is the inverse binomial transform of A000129(n-1). - Paul Barry (pbarry(AT)wit.ie), Apr 21 2004

Diagonal sums of A046854. a(n)=sum{k=0..n, binomial(floor(n/2), k)}. - Paul Barry (pbarry(AT)wit.ie), Jul 07 2004

a(n)=a(n-2)+2^floor((n-2)/2) - Paul Barry (pbarry(AT)wit.ie), Jul 14 2004

a(n)=sum{k=0..floor(n/2), binomial(floor(n/2), floor(k/2)) } - Paul Barry (pbarry(AT)wit.ie), Jul 15 2004

E.g.f.: cosh(asinh(1)+sqrt(2)*x)/sqrt(2). - Michael Somos Feb 28 2005

a(n)= Sum_{k, 0<=k<=n}A103633(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 03 2006

G.f.:(1+t)/(1-2*t^2). [From Dennis P. Walsh (dwalsh(AT)mtsu.edu), Oct 27 2009]

a(n)=2^(n/2)*((1+(-1)^n)/2+(1-(-1)^n)/(2*sqrt(2))). [From Paul Barry (pbarry(AT)wit.ie), Nov 12 2009]

Contribution from Dennis P. Walsh (dwalsh(AT)mtsu.edu), Oct 27 2009: (Start)

For n=5 the a(5)=4 symmetric subsets of [4] are {1,4}, {2,3}, {1,2,3,4} and

the empty set. (End)

seq(mul(mul(j, j=1..2), k=1..n/2), n=0..42); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 21 2007

a:=n->mul(numer(k/binomial(k, 2)), k=3..n): seq(a(n), n=2..44); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2008

seq(2^floor((n+1)/2), n=-1..41); [From Dennis P. Walsh (dwalsh(AT)mtsu.edu), Oct 27 2009]

Table[ 2^Floor[n/2], {n, 0, 42}] (from Robert G. Wilson v Jun 05 2004)

(PARI) a(n)=if(n<0, 0, 2^(n\2))

Cf. A006995, A057148.

Cf. A112030, A112033.

a(n) = A094718(3, n).

A001333 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 25 2009]

Sequence in context: A158780 A117575 A152166 this_sequence A060546 A163403 A120803

Adjacent sequences: A016113 A016114 A016115 this_sequence A016117 A016118 A016119

nonn,new

N. J. A. Sloane (njas(AT)research.att.com).