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A327665 Fibonacci with binary selection.

%I A327665 #46 Jan 20 2021 10:45:16
%S A327665 0,1,1,2,3,6,11,20,34,56,121,244,481,938,1832,3540,6757,12708,23410,
%T A327665 42328,73764,122889,275122,408967,832560,1580364,2834653,5044480,
%U A327665 8652446,13975133,29832886,58354102,112803422,215061934,401711254,737267883,1313954863,2259026414
%N A327665 Fibonacci with binary selection.
%H A327665 Hilarie Orman, <a href="/A327665/a327665.c.txt">C program for generating Fibonacci with binary selection</a>
%F A327665 a(n) = a(n-1) + Sum_{i=0..e(a(n-1))} b(a(n-1), e(a(n-1))-i)*a(n-i-2) where b(k, i) is the i-th bit in the binary expansion of k, with b(k, 0) being the low order bit of k, and e(k) = floor(log_2(k)). The initial terms are a(0) = 0, a(1) = 1. [edited by _Michel Marcus_, Sep 28 2019 and _Michael S. Branicky_, Jan 19 2021]
%t A327665 e[n_] := Floor[Log2[n]]; a[0] = 0; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 1] + Sum[BitGet[a[n - 1], em - i] * a[n - 2 - i], {i, 0, (em = e[a[n - 1]])}]; Array[a, 38, 0] (* _Amiram Eldar_, Sep 28 2019 *)
%o A327665 (PARI) lista(nn) = {my(va = vector(nn)); va[1] = 0; va[2] = 1; va[3] = 1; for (n=4, nn, my(b = binary(va[n-1])); va[n] = va[n-1] + sum(k=1, #b, b[k]*va[n-k-1]);); va;} \\ _Michel Marcus_, Sep 28 2019
%o A327665 (Python)
%o A327665 def aupton(nn):
%o A327665   alst = [0, 1, 1]
%o A327665   for n in range(3, nn+1):
%o A327665     b = list(map(int, bin(alst[n-1])[2:]))
%o A327665     alst.append(alst[n-1] + sum(bi*alst[n-i-2] for i, bi in enumerate(b)))
%o A327665   return alst[:nn+1]
%o A327665 print(aupton(37)) # _Michael S. Branicky_, Jan 19 2021
%Y A327665 Cf. A000045, A007088.
%K A327665 nonn,base
%O A327665 0,4
%A A327665 _Hilarie Orman_, Sep 21 2019