A003006 - OEIS

%I M0845 #40 Jan 05 2025 10:15:35

%S 1,1,2,3,7,15,35,81,195,473,1170,2920,7378,18787,48242,124658,324095,

%T 846872,2223352,5861011,15508423,41173560,109648734

%N Number of n-level ladder expressions with A001622.

%C 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

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

%H R. K. Guy and J. L. Selfridge, <a href="http://www.jstor.org/stable/2319392">The nesting and roosting habits of the laddered parenthesis</a>, Amer. Math. Monthly 80 (8) (1973), 868-876.

%H R. K. Guy and J. L. Selfridge, <a href="/A003018/a003018.pdf">The nesting and roosting habits of the laddered parenthesis</a>. (annotated cached copy)

%H <a href="/index/Par#parens">Index entries for sequences related to parenthesizing</a>

%t 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}]

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

%K nonn,more,hard

%O 1,3

%A _N. J. A. Sloane_

%E a(10)-a(23) added by _Vladimir Reshetnikov_, Mar 05 2019