crowlogic
commented
7 months ago A060632 a(n) = 2^wt(floor(n/2)) (i.e., 2^[A000120](https://oeis.org/A000120)(floor(n/2)), or [A001316](https://oeis.org/A001316)(floor(n/2))). 17
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 ([list](https://oeis.org/A060632/list); [graph](https://oeis.org/A060632/graph); [refs](https://oeis.org/search?q=A060632+-id:A060632); [listen](https://oeis.org/A060632/listen); [history](https://oeis.org/history?seq=A060632); [text](https://oeis.org/search?q=id:A060632&fmt=text); [internal format](https://oeis.org/A060632/internal))
OFFSET
0,3
COMMENTS
Number of conjugacy classes in the symmetric group S_n that have odd number of elements.
Also sequence [A001316](https://oeis.org/A001316) doubled.
Number of even numbers whose binary expansion is a child of the binary expansion of n. - [Nadia Heninger](https://oeis.org/wiki/User:Nadia_Heninger) and [N. J. A. Sloane](https://oeis.org/wiki/User:N._J._A._Sloane), Jun 06 2008
First differences of [A151566](https://oeis.org/A151566). Sequence gives number of toothpicks added at the n-th generation of the leftist toothpick sequence [A151566](https://oeis.org/A151566). - [N. J. A. Sloane](https://oeis.org/wiki/User:N._J._A._Sloane), Oct 20 2010
The Fi1 and Fi1 triangle sums, see [A180662](https://oeis.org/A180662) for their definitions, of Sierpiński's triangle [A047999](https://oeis.org/A047999) equal this sequence. - [Johannes W. Meijer](https://oeis.org/wiki/User: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](https://oeis.org/wiki/User: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](https://oeis.org/A060632/b060632.txt) (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](https://oeis.org/A000695/a000695_1.pdf), 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](https://scholarworks.calstate.edu/concern/theses/bk128j63v), 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](https://oeis.org/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS)
[Index entries for sequences related to toothpick sequences](https://oeis.org/index/To#toothpick)
FORMULA
a(n) = sum{k=0..floor(n/2), C(n, 2k) mod 2} - [Paul Barry](https://oeis.org/wiki/User:Paul_Barry), Jan 03 2005, Edited by [Harry J. Smith](https://oeis.org/wiki/User:Harry_J._Smith), Sep 15 2009
a(n) = gcd([A056040](https://oeis.org/A056040)(n), 2^n). - [Peter Luschny](https://oeis.org/wiki/User:Peter_Luschny), Jun 30 2011
G.f.: (1 + x) * Product_{k>=0} (1 + 2*x^(2^(k+1))). - [Ilya Gutkovskiy](https://oeis.org/wiki/User: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](https://oeis.org/wiki/User: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](https://oeis.org/A060632) := proc(n) local k; add(binomial(n, 2*k) mod 2, k=0..floor(n/2)); end: seq([A060632](https://oeis.org/A060632)(n), n=0..94); # edited by [Johannes W. Meijer](https://oeis.org/wiki/User:Johannes_W._Meijer), May 28 2011
[A060632](https://oeis.org/A060632) := n -> 2^add(i, i = convert(iquo(n, 2), base, 2)); # [Peter Luschny](https://oeis.org/wiki/User:Peter_Luschny), Jun 30 2011
[A060632](https://oeis.org/A060632) := n -> igcd(2^n, n! / iquo(n, 2)!^2); # [Peter Luschny](https://oeis.org/wiki/User:Peter_Luschny), Jun 30 2011
MATHEMATICA
a[n_] := 2^DigitCount[Floor[n/2], 2, 1]; Table[a[n], {n, 0, 94}] (* [Jean-François Alcover](https://oeis.org/wiki/User:Jean-Fran%C3%A7ois_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](https://oeis.org/wiki/User:Harry_J._Smith), Sep 14 2009
(PARI) a(n)=2^hammingweight(n\2) \\ [Charles R Greathouse IV](https://oeis.org/wiki/User: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](https://oeis.org/wiki/User:Klaus_Brockhaus), Oct 15 2010
(Python)
def [A060632](https://oeis.org/A060632)(n):
return 2**bin(n/2)[2:].count("1") # [Indranil Ghosh](https://oeis.org/wiki/User:Indranil_Ghosh), Feb 06 2017
CROSSREFS
Cf. [A000120](https://oeis.org/A000120), [A001316](https://oeis.org/A001316), [A139251](https://oeis.org/A139251), [A151566](https://oeis.org/A151566), [A160407](https://oeis.org/A160407).
Sequence in context: [A122386](https://oeis.org/A122386) [A051464](https://oeis.org/A051464) [A151565](https://oeis.org/A151565) * [A160407](https://oeis.org/A160407) [A007457](https://oeis.org/A007457) [A119802](https://oeis.org/A119802)
Adjacent sequences: [A060629](https://oeis.org/A060629) [A060630](https://oeis.org/A060630) [A060631](https://oeis.org/A060631) * [A060633](https://oeis.org/A060633) [A060634](https://oeis.org/A060634) [A060635](https://oeis.org/A060635)
KEYWORD
nonn
AUTHOR
Avi Peretz (njk(AT)netvision.net.il), Apr 15 2001
EXTENSIONS
More terms from [James A. Sellers](https://oeis.org/wiki/User:James_A._Sellers), Apr 16 2001
Edited by [N. J. A. Sloane](https://oeis.org/wiki/User:N._J._A._Sloane), Jun 06 2008; Oct 11 2010
a(0) = 1 added by [N. J. A. Sloane](https://oeis.org/wiki/User:N._J._A._Sloane), Sep 14 2009
Formula corrected by [Harry J. Smith](https://oeis.org/wiki/User:Harry_J._Smith), Sep 15 2009
STATUS
approved
Implement A060632 Sequence Class
Create
SymmetricGroupOddConjugacyClassCountsclass extendingIntegerSequence. It calculates the number of odd conjugacy classes in symmetric groups based on OEIS A060632. Details and sequence values are available on the OEIS entry.[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]