A003006 Number of n-level ladder expressions with A001622.

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

Author

N. J. A. Sloane

Keywords

• nonn
• more
• hard

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

• 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

Crossrefs

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

Programs

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}]

Extensions

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