A216763 The Lambda word generated by (1+sqrt(5))/2.

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, 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, 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

Offset: 0

Author

Norman Carey, Sep 15 2012

Keywords

• nonn

Comments

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.

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

Links

• N. Carey, On a class of locally symmetric sequences, The right infinite word Lambda Theta, in Mathematics and Computation in Music in Lect. Notes in Comp. Sci., Vol. 6726, Springer, (2011), 42-55.
• N. Carey, Lambda words: A class of rich words defined over an infinite alphabet, Journal of Integer Sequences, Vol. 16 (2013), Article 13.3.4.

Crossrefs

Cf. A216448, A216764.

Programs

Mathematica

t = GoldenRatio;
end = 100;
x = Table[Ceiling[n*1/t], {n, 0, end}];
y = Table[Ceiling[n*t], {n, 0, end}];
tot[p_, q_] := Total[Take[x, p + 1]] + (p*q) + Total[Take[y, q + 1]]
row[r_] := Table[tot[n, r], {n, 0, end - 1}]
g = Grid[Table[row[n], {n, 0, IntegerPart[(end - 1)/t]}]];
pos[n_] := Reverse[Position[g, n][[1, Range[2, 3]]] - 1]
d[n_] := (op[m_] := pos[m + 1] - pos[m];
  Abs[Total[ContinuedFraction[op[n][[1]]/op[n][[2]]]]])
l = Prepend[Table[d[n], {n, 1, 249}], 0]
(* Norman Carey, Sep 15 2012 *)