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 A263844 Constant term in expansion of n in Fraenkel's exotic ternary representation. 3
 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Let {p_i, i >= 0} = {1,3,7,17,41,99,...} denote the numerators of successive convergents to sqrt(2) (see A001333). Then any n >= 0 has a unique representation as n = Sum_{i >= 0} d_i*p_i, with 0 <= d_i <= 2, d_{i+1}=2 => d_i=0. Sequence gives a(n+1) = d_0. From Michel Dekking, Feb 11 2018: (Start) (a(n)) is the unique fixed point of the morphism alpha given by alpha: 0 -> 012, 1 -> 012, 2 -> 0. To see this, note first that the p_i satisfy p_{i+2}=2p_{i+1}+p_i for all i=1,2,... Then define a sequence of words by w(0) = 0, w(1) = 012, w(i+2) = w(i+1)w(i+1)w(i). The length of w(i) is equal to p_i. In the numeration system, the representation of n = p_i is d = 10..0, and the representation of n = 2p_i is d = 20..0. By unicity of the representation, the numbers n = p_i +m have the representation d = 1c, where c is the representation of m for m = 1,...,p_{i-1}. Similarly, because the digit 2 is required to be followed by the digit 0, the numbers n = 2p_i + m have the representation d =20c, where c is the representation of m for m = 1,...,p_{i-2}. It follows from this that the d_0 digits in the range 0 to p_{i+2} have to satisfy the equation w(i+2) = w(i+1)w(i+1)w(i). But alpha(0)=alpha(1)=w(1), and alpha(2)=w(0), which implies by induction that w(i) = alpha^i(0): w(i+1) = w(i)w(i)w(i-1) = alpha^i(0) alpha^i(0) alpha^{i-1}(0) = alpha^i(0) alpha^i(1)alpha^i(2) = alpha^i(012) = alpha^{i+1}(0). (a(n)) is modulo a change of alphabet (0->1, 1->2, 2->3) equal to A294180, the standard form of (a(n)). This combined with the fact that the Pell word A171588 is a Sturmian word leads to the formula for (a(n)) below. (End) LINKS Michel Dekking, Table of n, a(n) for n = 1..5000 Aviezri S. Fraenkel, On the recurrence f(m+1)= b(m)*f(m)-f(m-1) and applications, Discrete Mathematics 224 (2000), pp. 273-279. A. S. Fraenkel, An exotic ternary representation of the first few positive integers (Table 2 from Fraenkel (2000).) F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1. FORMULA a(n) = floor((n+2)r) + floor((n+1)r) - 2*floor(nr), where r = 1 - 1/sqrt(2). - Michel Dekking, Feb 11 2018 EXAMPLE See the link to Table 2 of Fraenkel (2000). MATHEMATICA Table[Floor[(n + 2) (1 - 1/Sqrt[2])] + Floor[(n + 1) (1 - 1/Sqrt[2])] - 2 Floor[n (1 - 1/Sqrt[2])], {n, 100}] (* Vincenzo Librandi, Feb 12 2018 *) PROG (Magma) [Floor((n+2)*(1-1/Sqrt(2)))+Floor((n+1)*(1-1/Sqrt(2)))- 2*Floor(n*(1-1/Sqrt(2))): n in [1..100]]; // Vincenzo Librandi, Feb 12 2018 CROSSREFS Cf. A001333, A270788. Sequence in context: A322989 A322823 A334872 * A079644 A072705 A072574 Adjacent sequences: A263841 A263842 A263843 * A263845 A263846 A263847 KEYWORD nonn AUTHOR N. J. A. Sloane, Nov 06 2015 EXTENSIONS More terms and new offset from Michel Dekking, Feb 11 2018 STATUS approved

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Last modified September 8 06:50 EDT 2024. Contains 375751 sequences. (Running on oeis4.)