%I A016116
%S A016116 1,1,2,2,4,4,8,8,16,16,32,32,64,64,128,128,256,256,512,512,1024,1024,
%T A016116 2048,2048,4096,4096,8192,8192,16384,16384,32768,32768,65536,65536,
%U A016116 131072,131072,262144,262144,524288,524288,1048576,1048576,2097152
%N A016116 2^[n/2].
%C A016116 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.
%C A016116 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
%C A016116 This sequence is the number of digits of each term of A061519. [From 
               Dmitry Kamenetsky (dkamen(AT)rsise.anu.edu.au), Jan 17 2009]
%C A016116 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]
%C A016116 Contribution from Dennis P. Walsh (dwalsh(AT)mtsu.edu), Oct 27 2009: 
               (Start)
%C A016116 a(n+1) is the number of symmetric subsets of [n]={1,2,...,n}. A subset 
               S of
%C A016116 [n] is symmetric if k is an element of S implies (n-k+1) is an element 
               of S. (End)
%D A016116 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal 
               Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%D A016116 E. Deutsch, Problem 1633, Math. Mag., 74 #5 (2001), p. 403.
%D A016116 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.
%H A016116 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A016116 S. Heubach and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0310197">
               Counting rises, levels and drops in compositions</a>
%H A016116 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=1067">
               Encyclopedia of Combinatorial Structures 1067</a>
%H A016116 Dennis Walsh, <a href="http://frank.mtsu.edu/~dwalsh/SYMMSUB.pdf">Notes 
               on symmetric subsets of {1, 2, ..., n}</a> [From Dennis P. Walsh 
               (dwalsh(AT)mtsu.edu), Oct 27 2009]
%F A016116 a(n) = a(n-1)*a(n-2)/a(n-3) = 2a(n-2) = 2^A004526(n). G.f.: (1+x)/(1-2x^2)
%F A016116 (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
%F A016116 E.g.f.: cosh(sqrt(2)x)+sinh(sqrt(2)x)/sqrt(2). - Paul Barry (pbarry(AT)wit.ie), 
               Jul 16 2003
%F A016116 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
%F A016116 Diagonal sums of A046854. a(n)=sum{k=0..n, binomial(floor(n/2), k)}. 
               - Paul Barry (pbarry(AT)wit.ie), Jul 07 2004
%F A016116 a(n)=a(n-2)+2^floor((n-2)/2) - Paul Barry (pbarry(AT)wit.ie), Jul 14 
               2004
%F A016116 a(n)=sum{k=0..floor(n/2), binomial(floor(n/2), floor(k/2)) } - Paul Barry 
               (pbarry(AT)wit.ie), Jul 15 2004
%F A016116 E.g.f.: cosh(asinh(1)+sqrt(2)*x)/sqrt(2). - Michael Somos Feb 28 2005
%F A016116 a(n)= Sum_{k, 0<=k<=n}A103633(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 03 2006
%F A016116 G.f.:(1+t)/(1-2*t^2). [From Dennis P. Walsh (dwalsh(AT)mtsu.edu), Oct 
               27 2009]
%F A016116 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]
%e A016116 Contribution from Dennis P. Walsh (dwalsh(AT)mtsu.edu), Oct 27 2009: 
               (Start)
%e A016116 For n=5 the a(5)=4 symmetric subsets of [4] are {1,4}, {2,3}, {1,2,3,
               4} and
%e A016116 the empty set. (End)
%p A016116 seq(mul(mul(j,j=1..2), k=1..n/2), n=0..42); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Sep 21 2007
%p A016116 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
%p A016116 seq(2^floor((n+1)/2),n=-1..41); [From Dennis P. Walsh (dwalsh(AT)mtsu.edu), 
               Oct 27 2009]
%t A016116 Table[ 2^Floor[n/2], {n, 0, 42}] (from Robert G. Wilson v Jun 05 2004)
%o A016116 (PARI) a(n)=if(n<0,0,2^(n\2))
%Y A016116 Cf. A006995, A057148.
%Y A016116 Cf. A112030, A112033.
%Y A016116 a(n) = A094718(3, n).
%Y A016116 A001333 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 25 2009]
%Y A016116 Sequence in context: A158780 A117575 A152166 this_sequence A060546 A163403 
               A120803
%Y A016116 Adjacent sequences: A016113 A016114 A016115 this_sequence A016117 A016118 
               A016119
%K A016116 nonn,new
%O A016116 0,3
%A A016116 N. J. A. Sloane (njas(AT)research.att.com).