A082392 Expansion of (1/x) * Sum_{k>=0} x^2^k / (1 - 2*x^2^(k+1)).

1, 1, 2, 1, 4, 2, 8, 1, 16, 4, 32, 2, 64, 8, 128, 1, 256, 16, 512, 4, 1024, 32, 2048, 2, 4096, 64, 8192, 8, 16384, 128, 32768, 1, 65536, 256, 131072, 16, 262144, 512, 524288, 4, 1048576, 1024, 2097152, 32, 4194304, 2048, 8388608, 2, 16777216

Offset: 0

Author

Ralf Stephan, Jun 07 2003

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..5000
• Ralf Stephan, Some divide-and-conquer sequences ...
• Ralf Stephan, Table of generating functions

Crossrefs

Cf. A045654, A220466.

Programs

Maple

nmax := 48: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := 2^n od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Feb 11 2013
A082392 := proc(n)
    2^A025480(n) ;
end proc:
seq(A082392(n),n=0..100) ; # R. J. Mathar, Jul 16 2020

Mathematica

a[n_] := 2^(((n+1)/2^IntegerExponent[n+1, 2]+1)/2-1);
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 15 2023 *)

PARI

for(n=0, 50, l=ceil(log(n+1)/log(2)); t=polcoeff(sum(k=0, l, (x^2^k)/(1-2*x^2^(k+1)))/x + O(x^(n+1)), n); print1(t", ");) ;

Formula

a(0) = 1, a(2*n) = 2^n, a(2*n+1) = a(n).

a(n) = 2^A025480(n) = 2^(A003602(n)-1).

a((2*n+1)*2^p-1) = 2^n, p >= 0 and n >= 0. - Johannes W. Meijer, Feb 11 2013