A216763 -id:A216763 - cp's OEIS frontend

A216448 The Lambda word generated by log(3)/log(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, 3, 5, 3, 3, 4, 3, 3, 5, 3, 3, 3, 5, 3, 3, 5, 3, 3, 3, 5, 3, 3, 5, 6, 5, 3, 3, 5, 3, 3, 5, 6, 5, 3, 3, 5, 6, 5, 3, 5, 6, 5, 3, 3, 5, 6, 5, 3, 5, 6, 5, 6, 5, 3, 5, 6, 5, 3, 5, 6, 5, 6, 5, 3, 5, 6, 5, 6, 5, 5, 6, 5, 6, 5, 3, 5, 6, 5, 6, 5, 5, 6

Offset: 0

Norman Carey, Sep 10 2012

nonn

A Lambda word is a symbolic sequence that encodes differences in the sequence i+jt, where t is irrational, 1 < t < 2.

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.

Cf. A003586, A216763, A216764.

t = Log[2, 3];
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_] := (d[0] = 0; 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 10 2012 *)

A216764 The Lambda word generated by Pi - 2.

Original entry on oeis.org

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

Norman Carey, Sep 15 2012

nonn

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

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.

Cf. A216763 (lambda word generated by (1+sqrt(5))/2), A216448 (lambda word generated by log(3)/log(2)).

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 *)

A239507 The Lambda word generated by E-1 (A091131).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 2, 4, 2, 2, 4, 5, 4, 2, 2, 4, 5, 4, 2, 4, 5, 4, 5, 4, 2, 4, 5, 4, 5, 4, 4, 5, 4, 5, 4, 5, 4, 4, 5, 4, 5, 4, 5, 6, 5, 4, 5, 4, 5, 4, 5, 6, 5, 4, 5, 4, 5, 6, 5, 5, 6, 5, 4, 5, 4, 5, 6, 5, 5, 6, 5, 4, 5, 6, 5, 5, 6, 5, 5, 6, 5, 4, 5, 6, 5, 5, 6, 5, 5, 6, 5, 6, 5, 5, 6, 5, 5, 6, 5, 5, 6, 5, 6, 5

Offset: 0

Norman Carey and Robert G. Wilson v, Mar 20 2014

nonn

A Lambda word is a symbolic sequence that encodes differences in the sequence i+j*t, where t is irrational, 1 < t < 2.

First occurrence of k>0: 1, 2, 5, 11, 19, 51, 119, 303, 571, 923, 1359, 4427, 10544, ..., .

Norman Carey and Robert G. Wilson v, Table of n, a(n) for n = 0..1024
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.

Cf. A216448, A216763, A216764.

t = E - 1; mx = 20; x = Table[ Ceiling[n*1/t], {n, 0, mx}]; y = Table[ Ceiling[n*t], {n, 0, mx}]; tot[p_, q_] := Total[ Take[x, p + 1]] + (p*q) + Total[ Take[y, q + 1]]; row[r_] := Table[ tot[n, r], {n, 0, mx - 1}]; g = Grid[ Table[ row[n], {n, 0, IntegerPart[(mx - 1)/t]}]]; pos[n_] := Reverse[ Position[ g, n][[1, Range[2, 3]]] - 1]; d[n_] := (d[0] = 0; op[m_] := pos[m + 1] - pos[m]; Abs[ Total[ ContinuedFraction[ op[n][[1]] / op[n][[2]] ]]]); lst = Prepend[ Table[ d[n], {n, 0, 249}], 0]

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A216448 The Lambda word generated by log(3)/log(2).

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A216764 The Lambda word generated by Pi - 2.

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A239507 The Lambda word generated by E-1 (A091131).

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