A263768 -id:A263768 - cp's OEIS frontend

A000016 a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.

1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94, 172, 316, 586, 1096, 2048, 3856, 7286, 13798, 26216, 49940, 95326, 182362, 349536, 671092, 1290556, 2485534, 4793492, 9256396, 17895736, 34636834, 67108864, 130150588, 252645136, 490853416

Offset: 0

N. J. A. Sloane

Also a(n+1) = number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the sum of its contents. E.g., for n=5 there are 6 such sequences.
Also a(n+1) = number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 0 (mod n+1) = size of Varshamov-Tenengolts code VT_0(n). E.g., |VT_0(5)| = 6 = a(6).
Number of binary necklaces with an odd number of zeros. - Joerg Arndt, Oct 26 2015
Also, number of subsets of {1,2,...,n-1} which sum to 0 modulo n (cf. A063776). - Max Alekseyev, Mar 26 2016
From Gus Wiseman, Sep 14 2019: (Start)
Also the number of subsets of {1..n} containing n whose mean is an element. For example, the a(1) = 1 through a(8) = 16 subsets are:
  1  2  3    4    5      6      7        8
        123  234  135    246    147      258
                  345    456    357      468
                  12345  1236   567      678
                         1456   2347     1348
                         23456  2567     1568
                                12467    3458
                                13457    3678
                                34567    12458
                                1234567  14578
                                         23578
                                         24568
                                         45678
                                         123468
                                         135678
                                         2345678
(End)
Number of self-dual binary necklaces with 2n beads (cf. A263768, A007147). - Bernd Mulansky, Apr 25 2023

			For n=3 the 2 output sequences are 000111000111... and 010101...
For n=5 the 4 output sequences are those with periodic parts {0000011111, 0001011101, 0010011011, 01}.
For n=6 there are 6 such sequences.

B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.
S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 172.
J. Hedetniemi and K. R. Hutson, Equilibrium of shortest path load in ring network, Congressus Numerant., 203 (2010), 75-95. See p. 83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, J. Dyn. Diff. Eqs. 20 (1) (2008) 201, eq. (39)

Seiichi Manyama, Table of n, a(n) for n = 0..3334 (first 201 terms from T. D. Noe)
Nicolás Álvarez, Victória Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, On extremal factors of de Bruijn-like graphs, Univ. Buenos Aires (Argentina 2023). See also arXiv:2308.16257 [math.CO], 2023. See references.
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 17.
A. E. Brouwer, The Enumeration of Locally Transitive Tournaments, Math. Centr. Report ZW138, Amsterdam, 1980.
S. Butenko, P. Pardalos, I. Sergienko, V. P. Shylo and P. Stetsyuk, Estimating the size of correcting codes using extremal graph problems, Optimization, 227-243, Springer Optim. Appl., 32, Springer, New York, 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Sébastien Designolle, Tamás Vértesi, and Sebastian Pokutta, Symmetric multipartite Bell inequalities via Frank-Wolfe algorithms, arXiv:2310.20677 [quant-ph], 2023.
T. M. A. Fink, Exact dynamics of the critical Kauffman model with connectivity one, arXiv:2302.05314 [cond-mat.stat-mech], 2023.
R. W. Hall and P. Klingsberg, Asymmetric rhythms and tiling canons, Amer. Math. Monthly, 113 (2006), 887-896.
A. A. Kulkarni, N. Kiyavash and R. Sreenivas, On the Varshamov-Tenengolts Construction on Binary Strings, 2013.
E. M. Palmer and R. W. Robinson, Enumeration of self-dual configurations, Pacific J. Math., 110 (1984), 203-221.
R. Pries and C. Weir, The Ekedahl-Oort type of Jacobians of Hermitian curves, arXiv preprint arXiv:1302.6261 [math.NT], 2013.
N. J. A. Sloane, On single-deletion-correcting codes
N. J. A. Sloane, Challenge Problems: Independent Sets in Graphs
Yan Bo Ti, Gabriel Verret, and Lukas Zobernig, Abelian Varieties with p-rank Zero, arXiv:2203.08401 [math.NT], 2022.
Antonio Vera López, Luis Martínez, Antonio Vera Pérez, Beatriz Vera Pérez, and Olga Basova, Combinatorics related to Higman's conjecture I: Parallelogramic digraphs and dispositions, Linear Algebra Appl. 530, 414-444 (2017).
Index entries for sequences related to tournaments
Index entries for sequences related to necklaces
Index entries for sequences related to subset sums modulo m

The main diagonal of table A068009, the left edge of triangle A053633.
Cf. A000048, A000031, A000013, A053634, A182469.
Subsets whose mean is an element are A065795.
Dominated by A082550.
Partitions containing their mean are A237984.
Subsets containing n but not their mean are A327477.
Cf. A051293, A135342, A240850, A327471, A327478.

a000016 0 = 1
a000016 n = (`div` (2 * n)) $ sum $
   zipWith (*) (map a000010 oddDivs) (map ((2 ^) . (div n)) $ oddDivs)
   where oddDivs = a182469_row n
-- Reinhard Zumkeller, May 01 2012

A000016 := proc(n) local d, t; if n = 0 then return 1 else t := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t := t + NumberTheory:-Totient(d)* 2^(n/d)/(2*n) fi od; return t fi end:

a[0] = 1; a[n_] := Sum[Mod[k, 2] EulerPhi[k]*2^(n/k)/(2*n), {k, Divisors[n]}]; Table[a[n], {n, 0, 35}](* Jean-François Alcover, Feb 17 2012, after Pari *)

a(n)=if(n<1,n >= 0,sumdiv(n,k,(k%2)*eulerphi(k)*2^(n/k))/(2*n));

from sympy import totient, divisors
def A000016(n): return sum(totient(d)<>(~n&n-1).bit_length(),generator=True))//n if n else 1 # Chai Wah Wu, Feb 21 2023

a(n) = Sum_{odd d divides n} (phi(d)*2^(n/d))/(2*n), n>0.
a(n) = A063776(n)/2.
a(n) = 2^(n-1) - A327477(n). - Gus Wiseman, Sep 14 2019

More terms from Michael Somos, Dec 11 1999

A007147 Number of self-dual 2-colored necklaces with 2n beads.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 9, 12, 23, 34, 63, 102, 190, 325, 612, 1088, 2056, 3771, 7155, 13364, 25482, 48175, 92205, 175792, 337594, 647326, 1246863, 2400842, 4636390, 8956060, 17334801, 33570816, 65108062, 126355336, 245492244, 477284182

Offset: 1

N. J. A. Sloane

For n>=4 also number of Napier cycle types for dimension d=n-3. See Böhm link. - Hugo Pfoertner, Oct 01 2013

Also the number of combinatorial types of simplicial neighborly polytopes in dimension 2n - 3 with 2n vertices. This sequence was described before the enumeration of self-dual necklaces: see references. See links for a bijection between the two objects. - Moritz Firsching, Aug 13 2015

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zhe Sun, T Suenaga, P Sarkar, S Sato, M Kotani, H Isobe, Stereoisomerism, crystal structures, and dynamics of belt-shaped cyclonaphthylenes, Proc. Nath. Acead. Sci. USA, vol. 113 no. 29, pp. 8109-8114, doi: 10.1073/pnas.1606530113

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Oswin Aichholzer and Anna Brötzner, Bicolored Order Types, Comp. Geom. Topology (2024) Vol. 3, No. 2, 3:1-3:17.
Amos Altshuler and Peter McMullen, The number of simplicial neighbourly d-polytopes with d + 3 vertices, Mathematika, 20(02):263-266, 1973., Theorem 1, p. 263.
Johannes Böhm, Generalized hyperbolic Napier cycles and their hyperbolic kernels, Part III, Jenaer Schriften zur Mathematik und Informatik, Math/inf/06/08, 2008.
Moritz Firsching, Realizability and inscribability for some simplicial spheres and matroid polytopes, , arXiv:1508.02531 [math.MG], 2015. See Appendix A1.
Bernd Mulansky and Andreas Potschka, A zonogon approach for computing small polygons of maximum perimeter, arXiv:2404.01841 [math.OC], 2024. See p. 9. See also Math. Program., 2025.
E. M. Palmer and R. W. Robinson, Enumeration of self-dual configurations, Pacific J. Math., 110 (1984), 203-221.
Index entries for sequences related to necklaces

Cf. A000016, A016116, A263768.

a[n_] := (1/2)*(2^Quotient[n-1, 2] + Total[(Mod[#, 2]*EulerPhi[#]*2^(n/#) & ) /@ Divisors[n]]/(2*n)); Table[a[n], {n, 1, 36}] (* Jean-François Alcover, Oct 24 2011, after Pari *)

a(n)= (1/2) *(2^((n-1)\2)+sumdiv(n,k,(k%2)*eulerphi(k)*2^(n/k))/(2*n))

def a(n):
    return 2^floor((n-3)/2)+1/(4*(n))*sum([euler_phi(h)*2^((n)/h) for h in divisors(n) if is_odd(h)])
# Moritz Firsching, Aug 13 2015

a(n) = (1/2) * (A016116(n-1) + A000016(n)).

a(n) = A263768(n) + 1. - Bernd Mulansky, Mar 08 2023

More terms from Michael Somos.

cp's OEIS Frontend

A000016 a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.

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