A003006 Number of n-level ladder expressions with A001622. (Formerly M0845)

1, 1, 2, 3, 7, 15, 35, 81, 195, 473, 1170, 2920, 7378, 18787, 48242, 124658, 324095, 846872, 2223352, 5861011, 15508423, 41173560, 109648734

Offset

1,3

Comments

Number of distinct values taken by phi^phi^...^phi (with n phi's and parentheses inserted in all possible ways), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vladimir Reshetnikov, Mar 05 2019

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..23.

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis. (annotated cached copy)

Index entries for sequences related to parenthesizing

Mathematica

ClearAll[phi, t, a]; t[1] = {0}; t[n_Integer] := t[n] = DeleteDuplicates[Flatten[Table[Outer[phi^#1 + #2 &, t[k], t[n - k]], {k, n - 1}]] /. phi^k_Integer :> Fibonacci[k] phi + Fibonacci[k - 1]]; a[n_Integer] := a[n] = Length[t[n]]; Table[a[n], {n, 23}]

Crossrefs

Cf. A001622, A002845, A082499, A052321 (first 10 terms match), A198683, A199812.

Sequence in context: A213920 A248869 A005909 * A052321 A114584 A328779

Adjacent sequences: A003003 A003004 A003005 * A003007 A003008 A003009

Keyword

nonn,more,hard

Author

N. J. A. Sloane

Extensions

a(10)-a(23) added by Vladimir Reshetnikov, Mar 05 2019

Status

approved