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 A016116 a(n) = 2^floor(n/2). 165
 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 compositions 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 Range of row n of the Circular Pascal array of order 4. - Shaun V. Ault, May 30 2014 a(n) is the number of permutations of length n avoiding both 213 and 312 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014 Also, the decimal representation of the diagonal from the origin to the corner (and from the corner to the origin except for the initial term) of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 190", based on the 5-celled von Neumann neighborhood when initialized with a single black (ON) cell at stage zero. - Robert Price, May 10 2017 a(n + 1) + n - 1, n > 0, is the number of maximal subsemigroups of the monoid of partial order-preserving or -reversing mappings on a set with n elements. See the East et al. link. - James Mitchell and Wilf A. Wilson, Jul 21 2017 Number of symmetric stairs with n cells. A stair is a snake polyomino allowing only two directions for adjacent cells: east and north. See A005418. - Christian Barrientos, May 11 2018 For n >= 4, a(n) is the exponent of the group of the Gaussian integers in a reduced system modulo (1+i)^(n+2). See A302254. - Jianing Song, Jun 27 2018 a(n) is the number of length-(n+1) binary sequences, denoted , with s(1)=1 and with s(i+1)=s(i) for odd i. - Dennis P. Walsh, Sep 06 2018 a(n+1) is the number of subsets of {1,2,..,n} in which all differences between successive elements of subsets are even.  For example, for n = 7, a(6) = 8 and the 8 subsets are {7}, {1,7}, {3,7}, {5,7}, {1,3,7}, {1,5,7}, {3,5,7}, {1,3,5,7}.  For odd differences between elements see Comment in A000045 (Fibonacci numbers). - Enrique Navarrete, Jul 01 2020 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..5000 Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics, Volume 332, 6 October 2014, Pages 45-54. Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, arXiv:1407.2197 [math.CO], 2014. Arvind Ayyer, Amritanshu Prasad, and Steven Spallone, Odd partitions in Young's lattice, arXiv:1601.01776 [math.CO], 2016. See Theorem 6 p. 12. Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4. Francesco Battistoni and Giuseppe Molteni, An elementary proof for a generalization of a Pohst's inequality, arXiv:2101.06163 [math.NT], 2021. Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015. Emeric Deutsch, Problem 1633, Math. Mag., 74 #5 (2001), p. 403. James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017. A. Goupil, H. Cloutier, and F. Nouboud, Enumeration of polyominoes inscribed in a rectangle Discrete Applied Mathematics 158(2010), 2014-2023. S. Heubach and T. Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1067 D. Levin, L. Pudwell, M. Riehl, and A. Sandberg, Pattern Avoidance on k-ary Heaps, Slides of Talk, 2014. 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. Agustín Moreno Cañadas, Hernán Giraldo, and Robinson Julian Serna Vanegas, Some integer partitions induced by orbits of Dynkin type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 12 (2017), pp. 2745-2766. 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] A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text). - N. J. A. Sloane, Mar 26 2015 Index entries for 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). a(n) = (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)^n*2^floor(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..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 a(n) = 2^((2*n - 1 + (-1)^n)/4). - Luce ETIENNE, Sep 20 2014 EXAMPLE For n=5 the a(5)=4 symmetric subsets of  are {1,4}, {2,3}, {1,2,3,4} and the empty set. - Dennis P. Walsh, Oct 27 2009 For n=5 the a(5)=4 length-6 binary sequences are <1,1,0,0,0,0>, <1,1,0,0,1,1>, <1,1,1,1,0,0> and <1,1,1,1,1,1>. - Dennis P. Walsh, Sep 06 2018 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}] (* Robert G. Wilson v, Jun 05 2004 *) With[{c=2^Range[0, 30]}, Riffle[c, c]] (* Harvey P. Dale, Jan 23 2015 *) CoefficientList[Series[(1+x)/(1-2*x^2), {x, 0, 50}], x] (* Stefano Spezia, Sep 07 2018 *) 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(); [next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013 (GAP) List([0..45], n->2^Int(n/2)); # Muniru A Asiru, Apr 03 2018 CROSSREFS Cf. A006995, A057148, A079944, A112030, A112033. a(n) = A094718(3, n). Cf. A001333. See A052955 for partial sums (without the initial term). A000079 gives the odd-indexed terms of a(n). Sequence in context: A131572 A152166 A320770 * A060546 A163403 A231208 Adjacent sequences:  A016113 A016114 A016115 * A016117 A016118 A016119 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Dec 11 1999 STATUS approved

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Last modified June 18 13:35 EDT 2021. Contains 345112 sequences. (Running on oeis4.)