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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).

%I A082388 #16 Sep 20 2019 08:55:21
%S A082388 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,
%T A082388 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,
%U A082388 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
%N 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).
%H A082388 Andrew Howroyd, <a href="/A082388/b082388.txt">Table of n, a(n) for n = 1..1024</a>
%F A082388 a(2^k) = 4*a(2^(k-1)) - 2*a(2^(k-2));
%F A082388 a(2^k) = round((1/2)*(2+sqrt(2))^k).
%F A082388 Multiplicative with a(2^e) = A006012(e), a(p^e) = 1 for odd prime p. - _Andrew Howroyd_, Jul 31 2018
%t A082388 a[n_] := With[{e = IntegerExponent[n, 2]}, Sum[Binomial[e, 2k] 2^(e-k), {k, 0, Quotient[e, 2]}]];
%t A082388 a /@ Range[1, 100] (* _Jean-François Alcover_, Sep 20 2019, from PARI *)
%o A082388 (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
%Y A082388 Cf. A006012.
%K A082388 nonn,mult
%O A082388 1,2
%A A082388 _Benoit Cloitre_, Apr 14 2003