This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A354773 #37 Nov 29 2023 13:08:09
%S A354773 0,2,0,1,2,3,8,0,1,5,12,2,3,7,16,8,9,0,10,1,4,11,24,12,13,2,14,3,6,15,
%T A354773 32,16,17,8,18,9,19,0,20,10,21,1,22,4,5,11,48,24,25,12,26,13,27,2,28,
%U A354773 14,29,3,30,6,7,15,64,32,33,16,34,17,35,8,36,18,37,9,38,19,39,0,40,20,41,10,42,21,43,1,44,22,45,4,46
%N A354773 For terms of A354169 that are the sum of two distinct powers of 2, the exponent of the smaller power of 2.
%H A354773 Rémy Sigrist, <a href="/A354773/b354773.txt">Table of n, a(n) for n = 1..10000</a>
%H A354773 Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, <a href="http://arxiv.org/abs/2209.04108">The Binary Two-Up Sequence</a>, arXiv:2209.04108 [math.CO], Sep 11 2022.
%H A354773 Rémy Sigrist, <a href="/A354773/a354773.txt">C++ program</a>
%F A354773 Conjecture from _N. J. A. Sloane_, Jun 29 2022: (Start)
%F A354773 The following is a conjectured recurrence for a(n). Basically a(n) = a(n/2-1) if n is even, and a(n) = (n+1)/2 if n is odd, except that there are four types of n which have a different formula, and there are 19 exceptional values for small n. Note that a(n) does not depend on earlier values when n is odd.
%F A354773 Here is the formula, which agrees with the first 10000 terms.
%F A354773 There are exceptional values as far out as n=61, so we take care of them first.
%F A354773 Initial conditons:
%F A354773 If n is on the list
%F A354773 [1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 21, 22, 29, 30, 45, 61]
%F A354773 then a(n) is given by the n-th term of the following list:
%F A354773 [0, 2, 0, 1, 2, 3, 8, 0, 1, 5, 12, 2, 3, 7, 16, 8, 9, 0, 10,
%F A354773 1, 4, 11, 24, 12, 13, 2, 14, 3, 6, 15, 32, 16, 17, 8, 18, 9,
%F A354773 19, 0, 20, 10, 21, 1, 22, 4, 5, 11, 48, 24, 25, 12, 26, 13,
%F A354773 27, 2, 28, 14, 29, 3, 30, 6, 7].
%F A354773 Otherwise, if n is even, a(n) = a(n/2-1).
%F A354773 Otherwise n is odd and is not one of the exceptions.
%F A354773 (I) If n = 3*2^k-3, k >= 5, then a(n) = (n-1)/4.
%F A354773 (II) If n = 2^k-3, k >= 4 then a(n) = (n-1)/4.
%F A354773 (III) If n = 3*2^k-1, k >= 2 then a(n) = n+1.
%F A354773 (IV) If n = 2^k-1, k >= 3 then a(n) = n+1.
%F A354773 (V) Otherwise a(n) = (n+1)/2.
%F A354773 (End)
%F A354773 The conjecture is now known to be true. See De Vlieger et al. (2022). - _N. J. A. Sloane_, Aug 29 2022
%o A354773 (Python)
%o A354773 from itertools import count, islice
%o A354773 from collections import deque
%o A354773 from functools import reduce
%o A354773 from operator import or_
%o A354773 def A354773_gen(): # generator of terms
%o A354773 aset, aqueue, b, f = {0,1,2}, deque([2]), 2, False
%o A354773 while True:
%o A354773 for k in count(1):
%o A354773 m, j, j2, r, s = 0, 0, 1, b, k
%o A354773 while r > 0:
%o A354773 r, q = divmod(r,2)
%o A354773 if not q:
%o A354773 s, y = divmod(s,2)
%o A354773 m += y*j2
%o A354773 j += 1
%o A354773 j2 *= 2
%o A354773 if s > 0:
%o A354773 m += s*2**b.bit_length()
%o A354773 if m not in aset:
%o A354773 if (s := bin(m)[:1:-1]).count('1') == 2:
%o A354773 yield s.index('1')
%o A354773 aset.add(m)
%o A354773 aqueue.append(m)
%o A354773 if f: aqueue.popleft()
%o A354773 b = reduce(or_,aqueue)
%o A354773 f = not f
%o A354773 break
%o A354773 A354773_list = list(islice(A354773_gen(),20)) # _Chai Wah Wu_, Jun 26 2022
%o A354773 (C++) See Links section.
%Y A354773 Cf. A354169, A354680, A354767, A354798, A354774, A354775.
%K A354773 base,nonn
%O A354773 1,2
%A A354773 _N. J. A. Sloane_, Jun 26 2022