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A285683 Positions of 2 in A285680.

%I A285683 #42 Mar 11 2025 02:55:02
%S A285683 5,10,12,17,22,24,29,31,36,41,43,48,53,55,60,62,67,72,74,79,81,86,91,
%T A285683 93,98,103,105,110,112,117,122,124,129,134,136,141,143,148,153,155,
%U A285683 160,162,167,172,174,179,184,186,191,193,198,203,205,210,212,217,222
%N A285683 Positions of 2 in A285680.
%C A285683 A 3-way partition of the positive integers, by positions of 0, 1, 2 in A285680:
%C A285683 A285681: positions of 0; slope t = (15+sqrt(5))/10;
%C A285683 A285682: positions of 1; slope u = 4+sqrt(5);
%C A285683 A285683: positions of 2; slope v = (1+3*sqrt(5))/2;
%C A285683 where 1/t + 1/u + 1/v = 1.
%C A285683 Conjecture: a(n) - a(n-1) is in {5,7} for n >= 2.
%C A285683 Proof of the correct version of Kimberling's conjecture: This conjecture is false; it should be replaced by a(n) - a(n-1) is in {2,5} for n >= 2. The truth of that conjecture follows from Lemma 8 in Allouche and Dekking, since (a(n)) is a generalized Beatty sequence (see Formula below). - _Michel Dekking_, Oct 09 2018
%C A285683 Apparently a(n) = A285679(n+1). - _R. J. Mathar_, May 14 2017
%C A285683 Proof of Mathar's conjecture: Let xF:=A003849 = 01001010010010100... be the Fibonacci sequence. The two return words of 1010 are r1=10100100, and r2=10100. [See Justin & Vuillon (2000) for definition of return word. - _N. J. A. Sloane_, Sep 23 2019] If these occur at index i, then in BOTH cases there is an occurrence of 0010 in xF at index i+3 in xF. These will be called type-A occurrences. Since r1 has suffix 00, there will also be occurrences of 0010 at index i+6, if r1 occurs at index i. These will be called type-B occurrences. It is clear that there are no other occurrences of 0010 in xF. Now if we perform the [0010->2] transform T on xF, then the type-B occurrences of 0010 will not yield the occurrence of a 2 in T(xF), since the StringReplace procedure acts from left to right, destroying the 0010 occurrences of type-B. The conclusion is that the positions of 2 in the [1010->2] transform and the [0010->2] transform are coupled, occurring at a fixed shift of each other, except that there is an extra first occurrence of 0010 at index 3. By coincidence, the first occurrence of 1010 in xF is at index 5, which equals the index of the second occurrence of 0010, with 3 subtracted (that has to be done because of the shrinking of a block of length 4 to the letter 2). It follows that the two occurrence sequences are equal, except for the extra 3. - _Michel Dekking_, Oct 09 2018
%H A285683 Clark Kimberling, <a href="/A285683/b285683.txt">Table of n, a(n) for n = 1..10000</a>
%H A285683 J.-P. Allouche and F. M. Dekking, <a href="https://arxiv.org/abs/1809.03424">Generalized Beatty sequences and complementary triples</a>, arXiv:1809.03424v3 [math.NT], 2018-2019.
%H A285683 Jacques Justin and Laurent Vuillon, <a href="http://www.numdam.org/item/ITA_2000__34_5_343_0/">Return words in Sturmian and episturmian words</a>, RAIRO-Theoretical Informatics and Applications 34.5 (2000): 343-356.
%F A285683 a(n) = 3*floor(n*phi) - n + 3 (this follows from Theorem 29 in Allouche and Dekking, since the overlap word 101010 that contains 1010 does not occur in the Fibonacci word). - _Michel Dekking_, Oct 09 2018
%t A285683 s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13] ; (* A003849 *)
%t A285683 w = StringJoin[Map[ToString, s]]
%t A285683 w1 = StringReplace[w, {"1010" -> "2"}]
%t A285683 st = ToCharacterCode[w1] - 48; (* A285680 *)
%t A285683 Flatten[Position[st, 0]];  (* A285681 *)
%t A285683 Flatten[Position[st, 1]];  (* A285682 *)
%t A285683 Flatten[Position[st, 2]];  (* A285683 *)
%Y A285683 Cf. A003849, A285680, A285681, A285682.
%K A285683 nonn,easy
%O A285683 1,1
%A A285683 _Clark Kimberling_, May 11 2017