OFFSET
0,2
COMMENTS
Bisection of A332840.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1502
FORMULA
a(n) = A332840(2*n).
a(n) = Product(A332758(k+2)) where k ranges over the positions of 1 bits in the binary expansion of n.
a(n) = big-Theta(C^n) for C = 4.63233857..., i.e., A*C^n < a(n) < B*C^n for constants A, B (but it's not the case that a(n) ~ C^n as lim inf a(n)/C^n and lim sup a(n)/C^n differ).
EXAMPLE
For n=1, the a(1)=3 fixed-point free involutions in a fixed Sylow 2-subgroup of S_4 (which subgroup is isomorphic to the dihedral group of degree 4) are (12)(34), (13)(24), and (14)(23).
MAPLE
b:= proc(n) b(n):=`if`(n=0, 0, b(n-1)^2+2^(2^(n-1)-1)) end:
a:= n-> (l-> mul(`if`(l[i]=1, b(i+1), 1), i=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..32); # Alois P. Heinz, Feb 27 2020
MATHEMATICA
a[n_] := Product[A332758[k+1], {k, Flatten@ Position[ Reverse@ IntegerDigits[n, 2], 1]}];
a /@ Range[0, 24] (* Jean-François Alcover, Apr 10 2020 *)
PROG
(PARI) a(n)={my(v=vector(logint(max(1, n), 2)+2)); v[1]=1; for(n=2, #v, v[n]=v[n-1]^2 + 2^(2^(n-1)-1)); prod(k=2, #v, if(bittest(n, k-2), v[k], 1))} \\ Andrew Howroyd, Feb 27 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Nick Krempel, Feb 27 2020
EXTENSIONS
Terms a(9) and beyond from Andrew Howroyd, Feb 27 2020
STATUS
approved