OFFSET
0,3
COMMENTS
Number of conjugacy classes in the symmetric group S_n that have odd number of elements.
Also sequence A001316 doubled.
Number of even numbers whose binary expansion is a child of the binary expansion of n. - Nadia Heninger and N. J. A. Sloane, Jun 06 2008
First differences of A151566. Sequence gives number of toothpicks added at the n-th generation of the leftist toothpick sequence A151566. - N. J. A. Sloane, Oct 20 2010
The Fi1 and Fi1 triangle sums, see A180662 for their definitions, of Sierpiński's triangle A047999 equal this sequence. - Johannes W. Meijer, Jun 05 2011
Also number of odd entries in n-th row of triangle of Stirling numbers of the first kind. - Istvan Mezo, Jul 21 2017
REFERENCES
I. G. MacDonald: Symmetric functions and Hall polynomials Oxford: Clarendon Press, 1979. Page 21.
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..65536 (terms 0..1000 from Harry J. Smith)
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Christina Talar Bekaroğlu, Analyzing Dynamics of Larger than Life: Impacts of Rule Parameters on the Evolution of a Bug's Geometry, Master's thesis, Calif. State Univ. Northridge (2023). See p. 92.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
FORMULA
a(n) = sum{k=0..floor(n/2), C(n, 2k) mod 2} - Paul Barry, Jan 03 2005, Edited by Harry J. Smith, Sep 15 2009
a(n) = gcd(A056040(n), 2^n). - Peter Luschny, Jun 30 2011
G.f.: (1 + x) * Product_{k>=0} (1 + 2*x^(2^(k+1))). - Ilya Gutkovskiy, Jul 19 2019
EXAMPLE
a(3) = 2 because in S_3 there are two conjugacy classes with odd number of elements, the trivial conjugacy class and the conjugacy class of transpositions consisting of 3 elements: (12),(13),(23).
From Omar E. Pol, Oct 12 2011 (Start):
Written as a triangle:
1,
1,
2,2,
2,2,4,4,
2,2,4,4,4,4,8,8,
2,2,4,4,4,4,8,8,4,4,8,8,8,8,16,16,
2,2,4,4,4,4,8,8,4,4,8,8,8,8,16,16,4,4,8,8,8,8,16,16,8,...
(End)
MAPLE
A060632 := proc(n) local k; add(binomial(n, 2*k) mod 2, k=0..floor(n/2)); end: seq(A060632(n), n=0..94); # edited by Johannes W. Meijer, May 28 2011
A060632 := n -> 2^add(i, i = convert(iquo(n, 2), base, 2)); # Peter Luschny, Jun 30 2011
A060632 := n -> igcd(2^n, n! / iquo(n, 2)!^2); # Peter Luschny, Jun 30 2011
MATHEMATICA
a[n_] := 2^DigitCount[Floor[n/2], 2, 1]; Table[a[n], {n, 0, 94}] (* Jean-François Alcover, Feb 25 2014 *)
PROG
(PARI) for (n=0, 1000, write("b060632.txt", n, " ", sum(k=0, floor(n/2), binomial(n, 2*k) % 2)) ) \\ Harry J. Smith, Sep 14 2009
(PARI) a(n)=2^hammingweight(n\2) \\ Charles R Greathouse IV, Feb 06 2017
(Magma) a000120:=func< n | &+Intseq(n, 2) >; [ 2^a000120(Floor(n/2)): n in [0..100] ]; // Klaus Brockhaus, Oct 15 2010
(Python)
def A060632(n):
return 2**bin(n/2)[2:].count("1") # Indranil Ghosh, Feb 06 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Avi Peretz (njk(AT)netvision.net.il), Apr 15 2001
EXTENSIONS
More terms from James A. Sellers, Apr 16 2001
Edited by N. J. A. Sloane, Jun 06 2008; Oct 11 2010
a(0) = 1 added by N. J. A. Sloane, Sep 14 2009
Formula corrected by Harry J. Smith, Sep 15 2009
STATUS
approved