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A355621 a(1) = 1; for n > 1, a(n) is the number of terms in the first n-1 terms of the sequence that share a 1-bit with a(n-1) in their binary expansions.

%I A355621 #16 Jul 11 2022 08:35:10
%S A355621 1,1,2,1,3,5,5,6,5,8,1,8,2,4,5,11,15,17,12,11,19,17,15,23,22,19,22,21,
%T A355621 24,16,10,18,20,21,29,33,22,30,33,23,38,31,42,28,35,37,38,37,40,22,41,
%U A355621 40,24,33,35,46,49,49,50,47,59,60,55,61,62,61,64,1,39,63,69,58,60,64,3,60,65,46,67
%N A355621 a(1) = 1; for n > 1, a(n) is the number of terms in the first n-1 terms of the sequence that share a 1-bit with a(n-1) in their binary expansions.
%C A355621 The indices where a(n) = 1 in the first 500000 terms are 1, 2, 4, 11, 68, 131, 2051, 4099. It is unknown if more exist. Many terms of the sequence are close to the line a(n) = n although only the first term is a possible fixed point. In the first 500000 terms the lowest values not to appear are 7, 9, 14, 25, 26. It is likely these and other numbers never appear although this is unknown.
%H A355621 Scott R. Shannon, <a href="/A355621/b355621.txt">Table of n, a(n) for n = 1..10000</a>
%H A355621 Scott R. Shannon, <a href="/A355621/a355621.png">Image of the first 500000 terms</a>. The green line is y = n.
%e A355621 a(7) = 5 as a(6) = 5 and the total number of terms in the first six terms that share a 1-bit with 5 in their binary expansions is five, namely 1, 1, 1, 3, 5.
%o A355621 (Python)
%o A355621 from itertools import count, islice
%o A355621 def agen():
%o A355621     an, alst = 1, [1]
%o A355621     for n in count(2):
%o A355621         yield an
%o A355621         an = sum(1 for k in alst if k&an)
%o A355621         alst.append(an)
%o A355621 print(list(islice(agen(), 79))) # _Michael S. Branicky_, Jul 10 2022
%Y A355621 Cf. A355625, A030190, A129760, A353989, A352763, A354606.
%K A355621 nonn,base
%O A355621 1,3
%A A355621 _Scott R. Shannon_, Jul 10 2022