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A342637 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 8*a(n) + a(n+1).

%I A342637 #11 Mar 30 2021 20:09:20
%S A342637 0,1,1,9,1,17,9,73,1,25,17,145,9,145,73,585,1,33,25,217,17,281,145,
%T A342637 1169,9,217,145,1233,73,1169,585,4681,1,41,33,289,25,417,217,1753,17,
%U A342637 417,281,2393,145,2329,1169,9361,9,289,217,1881,145,2393,1233,9937,73,1753,1169,9937,585
%N A342637 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 8*a(n) + a(n+1).
%H A342637 Alois P. Heinz, <a href="/A342637/b342637.txt">Table of n, a(n) for n = 0..16384</a>
%F A342637 G.f.: x * Product_{k>=0} (1 + x^(2^k) + 8*x^(2^(k+1))).
%F A342637 a(n) == 1 (mod 8) for n >= 1. - _Hugo Pfoertner_, Mar 17 2021
%p A342637 a:= proc(n) option remember; `if`(n<2, n, (q->
%p A342637      `if`(d=1, 8*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
%p A342637     end:
%p A342637 seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 17 2021
%t A342637 a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 8 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 60}]
%t A342637 nmax = 60; CoefficientList[Series[x Product[(1 + x^(2^k) + 8 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
%Y A342637 Cf. A002487, A116528, A178243, A342603, A342633, A342634, A342635, A342636, A342638.
%K A342637 nonn
%O A342637 0,4
%A A342637 _Ilya Gutkovskiy_, Mar 17 2021