A060632 a(n) = 2^wt(floor(n/2)) (i.e., 2^A000120(floor(n/2)), or A001316(floor(n/2))).

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, 8, 16, 16, 16, 16, 32, 32, 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, 8, 16, 16, 16, 16, 32

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

Index entries for sequences related to toothpick sequences

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

Cf. A000120, A001316, A139251, A151566, A160407.

Sequence in context: A122386 A051464 A151565 * A160407 A007457 A119802

Adjacent sequences: A060629 A060630 A060631 * A060633 A060634 A060635

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