A216764 The Lambda word generated by Pi - 2.

0, 1, 2, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 9, 8, 1, 1, 1, 1, 1, 1, 8, 9, 8, 9, 8, 1, 1, 1, 1, 1, 8, 9, 8, 9, 8, 9, 8, 1, 1, 1, 1, 8, 9, 8, 9, 8, 9, 8, 9, 8, 1, 1, 1, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 8, 1, 1

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. Here, t = Pi - 2.

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

First occurrence of k>0: 2, 3, 6, 10, 15, 21, 28, 36, 45, 144, 299, 510, 777, 1100, 1479, ..., . - Robert G. Wilson v, Feb 25 2017

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..1000
• 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. A216763 (lambda word generated by (1+sqrt(5))/2), A216448 (lambda word generated by log(3)/log(2)).

Programs

Mathematica

t = Pi - 2;
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 16 2012 *)