A216763 - OEIS

%I #22 Mar 24 2014 02:09:23

%S 0,1,2,1,2,3,2,2,3,2,3,4,3,2,3,4,3,3,4,3,4,3,3,4,3,4,5,4,3,4,3,4,5,4,

%T 3,4,5,4,4,5,4,3,4,5,4,4,5,4,5,4,4,5,4,4,5,4,5,4,4,5,4,5,6,5,4,5,4,4,

%U 5,4,5,6,5,4,5,4,5,6,5,4,5,6,5,4,5,4,5,6,5,4,5,6,5,5,6,5,4,5,6

%N The Lambda word generated by (1+sqrt(5))/2.

%C A Lambda word is a symbolic sequence that encodes differences in the sequence i+j*t, where t is irrational, 1 < t < 2. This is the Fibonacci Lambda word, t = (1+sqrt(5))/2. The word is achieved by connecting the position numbers of the integers in order from a transpose of the array form of A216448 (0,0), (1,0), (0,1), (2,0), (1,1), and then encoding the vectors starting with (1,0) -> 0, (-1,1) -> 1, (2,-1) -> 2, (-1,1) -> 1.

%C A Lambda word is a right infinite rich word on an infinite alphabet.

%H N. Carey, <a href="http://dx.doi.org/10.1007/978-3-642-21590-2_4">On a class of locally symmetric sequences, The right infinite word Lambda Theta</a>, in Mathematics and Computation in Music in Lect. Notes in Comp. Sci., Vol. 6726, Springer, (2011), 42-55.

%H N. Carey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Carey/carey6.html">Lambda words: A class of rich words defined over an infinite alphabet</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.3.4.

%t t = GoldenRatio;

%t end = 100;

%t x = Table[Ceiling[n*1/t], {n, 0, end}];

%t y = Table[Ceiling[n*t], {n, 0, end}];

%t tot[p_, q_] := Total[Take[x, p + 1]] + (p*q) + Total[Take[y, q + 1]]

%t row[r_] := Table[tot[n, r], {n, 0, end - 1}]

%t g = Grid[Table[row[n], {n, 0, IntegerPart[(end - 1)/t]}]];

%t pos[n_] := Reverse[Position[g, n][[1, Range[2, 3]]] - 1]

%t d[n_] := (op[m_] := pos[m + 1] - pos[m];

%t   Abs[Total[ContinuedFraction[op[n][[1]]/op[n][[2]]]]])

%t l = Prepend[Table[d[n], {n, 1, 249}], 0]

%t (* _Norman Carey_, Sep 15 2012 *)

%Y Cf. A216448, A216764.

%K nonn

%O 0,3

%A _Norman Carey_, Sep 15 2012