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

2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, 49, 52, 54, 57, 60, 62, 65, 68, 70, 73, 75, 78, 81, 83, 86, 89, 91, 94, 96, 99, 102, 104, 107, 109, 112, 115, 117, 120, 123, 125, 128, 130, 133, 136, 138, 141, 143, 146, 149, 151, 154, 157, 159, 162

Offset: 1

Author

Amarnath Murthy, Dec 14 2003

Keywords

• nonn

Comments

Is this the same as A001950? - Alec Mihailovs (alec(AT)mihailovs.com), Jul 23 2007

Identical to n + A066096(n)? - Ed Russell (times145(AT)hotmail.com), May 09 2009

From Michel Dekking, Dec 18 2024: (Start)

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.

The answer to Russell’s question is well-known, and also Detlef’s formula is well-known.

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)

Examples

			A073869 = A005206 = 0,1,1,2,3,3,4,4,5,6,6,... The pair (1,1) occurs at k = 2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• F. M. Dekking, On Hofstadter's G-sequence, Journal of Integer Sequences 26 (2023), Article 23.9.2, 1-11.

Crossrefs

Cf. A001622, A001950, A002251, A005206, A073869, A090908.

Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.

Cf. A004926, A004927, A004928, A004929, A004930, A004931, A004932.

Cf. A004933, A004934, A004935, A004976, A066096.

Programs

Magma

[Floor((3+Sqrt(5))*n/2): n in [0..80]]; // G. C. Greubel, Sep 12 2023

Mathematica

(* First program *)
A002251= Fold[Append[#1, #2 Ceiling[#2/GoldenRatio] -Total[#1]] &, {1}, Range[2, 500]] - 1; (* Birkas Gyorgy's code of A019444, modified *)
A090909= Join[{0}, Select[Partition[A002251, 2, 1], #[[2]] > #[[1]] &][[All, 2]]] (* G. C. Greubel, Sep 12 2023 *)
(* Second program *)
Floor[GoldenRatio^2*Range[0,80]] (* G. C. Greubel, Sep 12 2023 *)

SageMath

[floor(golden_ratio^2*n) for n in range(81)] # G. C. Greubel, Sep 12 2023

Formula

a(n) = floor(phi^2*n), where phi = (1+sqrt(5))/2. - Gary Detlefs, Mar 10 2011

Extensions

More terms from R. J. Mathar, Sep 29 2017

Name corrected by Michel Dekking, Dec 13 2024