A082388 a(1) = 1, a(2) = 2; further terms are defined by rules that for k >= 2, a(2^k-i) = a(2^k+i) for 1 <= i <= 2^k-1 and a(2^k) = a(2^(k-1)) + Sum_{i < 2^k} a(i).

1, 2, 1, 6, 1, 2, 1, 20, 1, 2, 1, 6, 1, 2, 1, 68, 1, 2, 1, 6, 1, 2, 1, 20, 1, 2, 1, 6, 1, 2, 1, 232, 1, 2, 1, 6, 1, 2, 1, 20, 1, 2, 1, 6, 1, 2, 1, 68, 1, 2, 1, 6, 1, 2, 1, 20, 1, 2, 1, 6, 1, 2, 1, 792, 1, 2, 1, 6, 1, 2, 1, 20, 1, 2, 1, 6, 1, 2, 1, 68, 1, 2, 1, 6, 1, 2, 1, 20, 1, 2, 1, 6, 1, 2, 1, 232, 1, 2, 1

Offset: 1

Author

Benoit Cloitre, Apr 14 2003

Keywords

• nonn
• mult

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1024

Crossrefs

Cf. A006012.

Programs

Mathematica

a[n_] := With[{e = IntegerExponent[n, 2]}, Sum[Binomial[e, 2k] 2^(e-k), {k, 0, Quotient[e, 2]}]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019, from PARI *)

PARI

a(n)={my(e=valuation(n,2)); sum(k=0, e\2, binomial(e, 2*k)*2^(e-k))} \\ Andrew Howroyd, Jul 31 2018

Formula

a(2^k) = 4*a(2^(k-1)) - 2*a(2^(k-2));

a(2^k) = round((1/2)*(2+sqrt(2))^k).

Multiplicative with a(2^e) = A006012(e), a(p^e) = 1 for odd prime p. - Andrew Howroyd, Jul 31 2018