A007238 Length of longest chain of subgroups in S_n.
0, 1, 2, 4, 5, 6, 7, 10, 11, 12, 13, 15, 16, 17, 18, 22, 23, 24, 25, 27, 28, 29, 30, 33, 34, 35, 36, 38, 39, 40, 41, 46, 47, 48, 49, 51, 52, 53, 54, 57, 58, 59, 60, 62, 63, 64, 65, 69, 70, 71, 72, 74, 75, 76, 77, 80, 81, 82, 83, 85, 86, 87, 88, 94, 95, 96, 97, 99, 100, 101
Offset: 1
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- T. D. Noe, Table of n, a(n) for n=1..1000
- J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
- J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II
- L. Babai, On the length of subgroup chains in the symmetric group, Commun. Algebra, 14 (1986), 1729-1736.
- P. J. Cameron, M. Gadouleau, J. D. Mitchell, Y. Peresse, Chains of subsemigroups, arXiv preprint arXiv:1501.06394 [math.GR], 2015.
- Peter J. Cameron; Ron Solomon; Alexandre Turull, Chains of subgroups in symmetric groups, J. Algebra 127 (1989), no. 2, 340-352.
- Donald M. Davis, Divisibility by 2 and 3 of certain Stirling numbers, arXiv:0807.2629 [math.NT], Jul 16, 2008.
Programs
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Maple
A000120 := proc(n) convert(n,base,2) ; add(i,i=%) ; end proc: A007238 := proc(n) floor((3*n-1)/2)-A000120(n) ; end proc: seq(A007238(n),n=1..20) ;
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Mathematica
a[n_] := Ceiling[ 3n/2 ] - Count[ IntegerDigits[n, 2], 1] - 1; Table[ a[n], {n, 1, 70}] (* Jean-François Alcover, Jan 19 2012, after formula *) Table[Ceiling[(3n)/2]-DigitCount[n,2,1]-1,{n,70}] (* Harvey P. Dale, Nov 20 2021 *) -
PARI
vector(70, n, ceil(3*n/2) - hammingweight(n) - 1) \\ Joerg Arndt, May 16 2016
Formula
a(n) = ceiling(3n/2) - b(n) - 1, where b(n) = # 1's in binary expansion of n (A000120).
G.f.: 1/(1-x) * (-1/(1-x^2) + Sum(k>=0, x^2^k/(1-x^2^k))). - Ralf Stephan, Apr 13 2002
Comments