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A016116 2^floor(n/2). 131
1, 1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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, Dec 10 2001

This sequence is the number of digits of each term of A061519. - Dmitry Kamenetsky, 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,...). - Gary W. Adamson, Mar 25 2009

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. - Dennis P. Walsh, Oct 27 2009

INVERT and inverse INVERT transforms give A006138, A039834(n-1).

The Kn21 sums, see A180662, of triangle A065941 equal the terms of this sequence. - Johannes W. Meijer, Aug 15 2011]

First differences of A027383. - Jason Kimberley, Nov 01 2011

Run lengths in A079944. - Jeremy Gardiner, Nov 21 2011

Number of binary palindromes (A006995) between 2^(n-1) and 2^n (for n>1). - Hieronymus Fischer, Feb 17 2012

Pisano period lengths: 1, 1, 4, 1, 8, 4, 6, 1, 12, 8, 20, 4, 24, 6, 8, 1, 16, 12, 36, 8,... - R. J. Mathar, Aug 10 2012

REFERENCES

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

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

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

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1067

Laurent Noé, Spaced seed design on profile HMMs for precise HTS read-mapping efficient sliding window product on the matrix semi-group, in Rapide Bilan 2012-2013, Laurent, LIFL, Université Lille 1 - INRIA Journées au vert 11 et 12 juin 2013, Laurent, Année 2012-2013.

Valentin Ovsienko, Villes paires et impaires (Oddtown and Eventown) I, Images des Mathématiques, CNRS, 2013 (in French).

Dennis Walsh, Notes on symmetric subsets of {1, 2, ..., n} [From Dennis P. Walsh, Oct 27 2009]

Index to divisibility sequences

Index entries for sequences related to linear recurrences with constant coefficients, signature (0,2).

FORMULA

a(n) = a(n-1)*a(n-2)/a(n-3) = 2*a(n-2) = 2^A004526(n).

G.f.: (1+x)/(1-2*x^2)

(1/2+sqrt(1/8))*sqrt(2)^n+(1/2-sqrt(1/8))*(-sqrt(2))^n. - Ralf Stephan, Mar 11 2003

E.g.f.: cosh(sqrt(2)*x)+sinh(sqrt(2)*x)/sqrt(2). - Paul Barry, 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, Apr 21 2004

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

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

a(n) = sum{k=0..floor(n/2), binomial(floor(n/2), floor(k/2)) }. - Paul Barry, 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 Deléham, Dec 03 2006

a(n) = 2^(n/2)*((1+(-1)^n)/2+(1-(-1)^n)/(2*sqrt(2))). - Paul Barry, Nov 12 2009

EXAMPLE

For n=5 the a(5)=4 symmetric subsets of [4] are {1,4}, {2,3}, {1,2,3,4} and the empty set. - Dennis P. Walsh, Oct 27 2009

MAPLE

A016116:= proc(n): 2^floor(n/2) end: seq(A016116(n), n=0..42); - Dennis P. Walsh, Oct 27 2009

MATHEMATICA

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

PROG

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

(MAGMA) [2^Floor(n/2): n in [0..50]]; // Vincenzo Librandi, Aug 16 2011

(Maxima) makelist(2^floor(n/2), n, 0, 50); [Martin Ettl, Oct 17 2012]

(Sage)

def A016116():

    x, y = -1, 0

    while true:

        yield -x

        x, y = x + y, x - y

a = A016116(); [a.next() for i in range(40)]  # Peter Luschny, Jul 11 2013

CROSSREFS

Cf. A006995, A057148, A079944, A112030, A112033.

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

Cf. A001333.

See A052955 for partial sums (without the initial term).

Sequence in context: A117575 A131572 A152166 * A060546 A163403 A231208

Adjacent sequences:  A016113 A016114 A016115 * A016117 A016118 A016119

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 19 00:07 EDT 2014. Contains 240735 sequences.