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A090909 Terms a(k) of A073869 for which a(k-1) = a(k), and a(k) and a(k+1) are distinct.

%I A090909 #46 Dec 19 2024 16:20:31
%S A090909 2,5,7,10,13,15,18,20,23,26,28,31,34,36,39,41,44,47,49,52,54,57,60,62,
%T A090909 65,68,70,73,75,78,81,83,86,89,91,94,96,99,102,104,107,109,112,115,
%U A090909 117,120,123,125,128,130,133,136,138,141,143,146,149,151,154,157,159,162
%N A090909 Terms a(k) of A073869 for which a(k-1) = a(k), and a(k) and a(k+1) are distinct.
%C A090909 Is this the same as A001950? - Alec Mihailovs (alec(AT)mihailovs.com), Jul 23 2007
%C A090909 Identical to n + A066096(n)? - Ed Russell (times145(AT)hotmail.com), May 09 2009
%C A090909 From _Michel Dekking_, Dec 18 2024: (Start)
%C A090909 Proof of Mihailovs's conjecture: This follows immediately from the result in my 2023 paper in JIS that A073869 is equal to Hofstadter’s G-sequence A005206, and my recent comment in A005206 on the pairs of duplicate values in A005206.
%C A090909 The answer to Russell’s question is well-known, and also Detlef’s formula is well-known.
%C A090909 Originally, this sequence was given the name “Terms a(k) of A073869 for which a(k-1), a(k) and a(k+1) are distinct.’’ These are the triples (1,2,3),(4,5,6),(6,7,8), (9,10,11), ... occurring at  k = 3, k = 8, k = 11, k = 16,... in A005206. Note that if a duplicate pair (a(m-1), a(m)) is followed directly by another duplicate pair, then a(m-3), a(m-2) and a(m-1) are distinct, and only so. This corresponds to the block 00 occurring in the Fibonacci word obtained by projecting A005206 on the Fibonacci word (see Corollary in my recent comment in A005206). These occurrences are at the Wythoff AB numbers A003623 according to Wolfdieter Lang’s comment in A003623. Conclusion: the sequence of terms a(k) of A073869 for which a(k-1), a(k), and a(k+1) are distinct is given by the Wythoff AB-numbers. (End)
%H A090909 G. C. Greubel, <a href="/A090909/b090909.txt">Table of n, a(n) for n = 0..10000</a>
%H A090909 F. M. Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Dekking2/dek9.html">On Hofstadter's G-sequence</a>, Journal of Integer Sequences 26 (2023), Article 23.9.2, 1-11.
%F A090909 a(n) = floor(phi^2*n), where phi = (1+sqrt(5))/2. - _Gary Detlefs_, Mar 10 2011
%e A090909 A073869 = A005206 = 0,1,1,2,3,3,4,4,5,6,6,... The pair (1,1) occurs at k = 2.
%t A090909 (* First program *)
%t A090909 A002251= Fold[Append[#1, #2 Ceiling[#2/GoldenRatio] -Total[#1]] &, {1}, Range[2, 500]] - 1; (* Birkas Gyorgy's code of A019444, modified *)
%t A090909 A090909= Join[{0}, Select[Partition[A002251, 2, 1], #[[2]] > #[[1]] &][[All, 2]]] (* _G. C. Greubel_, Sep 12 2023 *)
%t A090909 (* Second program *)
%t A090909 Floor[GoldenRatio^2*Range[0,80]] (* _G. C. Greubel_, Sep 12 2023 *)
%o A090909 (Magma) [Floor((3+Sqrt(5))*n/2): n in [0..80]]; // _G. C. Greubel_, Sep 12 2023
%o A090909 (SageMath) [floor(golden_ratio^2*n) for n in range(81)] # _G. C. Greubel_, Sep 12 2023
%Y A090909 Cf. A001622, A001950, A002251, A005206, A073869, A090908.
%Y A090909 Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
%Y A090909 Cf. A004926, A004927, A004928, A004929, A004930, A004931, A004932.
%Y A090909 Cf. A004933, A004934, A004935, A004976, A066096.
%K A090909 nonn
%O A090909 1,1
%A A090909 _Amarnath Murthy_, Dec 14 2003
%E A090909 More terms from _R. J. Mathar_, Sep 29 2017
%E A090909 Name corrected by _Michel Dekking_, Dec 13 2024