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A372398 Numbers occurring exactly twice in Hofstadter G/H-like sequence H_5 (A005376).

%I A372398 #25 Jan 05 2025 19:51:42
%S A372398 1,6,7,9,12,16,21,26,27,32,33,35,40,41,43,46,51,52,54,57,61,66,67,69,
%T A372398 72,76,81,86,87,89,92,96,101,106,107,112,113,115,118,122,127,132,133,
%U A372398 138,139,141,146,147,149,152,156,161,166,167,172,173,175,180,181
%N A372398 Numbers occurring exactly twice in Hofstadter G/H-like sequence H_5 (A005376).
%C A372398 Also first prepending column of the 5-Zeckendorf array (see Ericksen and Anderson).
%C A372398 _N. J. A. Sloane_ observed already the relation between Hofstadter G/H-like sequences H_k and k-Zeckendorf arrays in May 2003, at least for k = 3 (see formula section and history of A005374). First observation most probably by Diego Torres, Nov 2002, relating the Hofstadter G/H-like sequences H_k with the k-Zeckendorf arrays and Lamé sequences of order k (see comments in A005375 and A005376).
%H A372398 A.H.M. Smeets, <a href="/A372398/b372398.txt">Table of n, a(n) for n = 1..20000</a>
%H A372398 Larry Ericksen and Peter G. Anderson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/50-1/EricksenAnderson.pdf">Patterns in differences between rows in k-Zeckendorf arrays</a>, The Fibonacci Quarterly, Vol. 50, February 2012.
%o A372398 (Python)
%o A372398 def H(n,k):
%o A372398     if n == 0:
%o A372398         return 0
%o A372398     else:
%o A372398         i, x = 0, n-1
%o A372398         while i < k:
%o A372398             i, x = i+1, H(x,k)
%o A372398         return n-x
%o A372398 n, nn = 0, 0
%o A372398 while n < 59:
%o A372398     if nn == 0:
%o A372398         Hno = H(nn,5)
%o A372398     else:
%o A372398         Hnn = H(nn,5)
%o A372398         if Hnn == Hno:
%o A372398             n += 1
%o A372398             print(Hnn, end = ", ")
%o A372398         Hno = Hnn
%o A372398     nn += 1
%Y A372398 Cf. A005374, A005376.
%Y A372398 Numbers occurring exactly twice in Hofstadter G/H like sequence H_k: A000291 (k=2), A005374 (k=3), A372397 (k=4), this sequence (k=5).
%K A372398 nonn
%O A372398 1,2
%A A372398 _A.H.M. Smeets_, Apr 29 2024