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A352228 Number of length-n blocks in the Thue-Morse sequence with intertwining pattern ABBA ABBA ABBA... .

%I A352228 #27 Dec 25 2024 04:03:31
%S A352228 0,0,1,1,2,2,2,3,4,4,4,4,4,5,6,7,8,8,8,8,8,8,8,8,8,9,10,11,12,13,14,
%T A352228 15,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,17,18,19,20,21,
%U A352228 22,23,24,25,26,27,28,29,30,31,32,32,32,32,32,32,32,32,32,32,32,32,32,32
%N A352228 Number of length-n blocks in the Thue-Morse sequence with intertwining pattern ABBA ABBA ABBA... .
%C A352228 Essentially a duplicate of A060973. This is 0 together with A060973.
%C A352228 The intertwining pattern is the list of consecutive occurrences of a block x and its binary complement x' in the Thue-Morse sequence A010060, where A codes an occurrence of x and B codes an occurrence of x'.
%H A352228 Jeffrey Shallit, <a href="https://arxiv.org/abs/2203.02917">Intertwining of Complementary Thue-Morse Factors</a>, arxiv preprint arXiv:2203.02917 [cs.FL], March 6 2022.
%F A352228 a(n) = A060973(n-1) for n >= 1.
%e A352228 For n = 4, the only block with intertwining sequence ABBA ABBA ... is 0011.
%t A352228 a[n_] := a[n] = Switch[n, 1|2, 0, 3, 1, n, If[Mod[n, 2] == 1, 2*a[(n+1)/2//Floor], a[n/2//Floor] + a[1+n/2//Floor]]];
%t A352228 Table[a[n], {n, 1, 78}] (* _Jean-François Alcover_, Mar 25 2023 *)
%o A352228 (Python) # Recurrence from _Henry Bottomley_ in A060973.
%o A352228 from functools import cache
%o A352228 @cache
%o A352228 def a(n):
%o A352228     match n:
%o A352228         case 1 | 2: return 0
%o A352228         case 3: return 1
%o A352228         case n  if n % 2 == 1: return 2*a((n+1)//2)
%o A352228         case _: return a(n//2) + a(1+n//2)
%o A352228 print([a(n) for n in range(1, 73)])  # _Peter Luschny_, Mar 08 2022
%Y A352228 Cf. A010060, A060973. Related to A352227.
%K A352228 nonn
%O A352228 1,5
%A A352228 _Jeffrey Shallit_, Mar 08 2022