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A217710 Cardinality of the set of possible heights of AVL trees with n (leaf-) nodes.

%I A217710 #39 Aug 14 2021 18:34:24
%S A217710 1,1,1,1,1,1,1,2,1,1,1,1,2,2,2,2,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,1,2,
%T A217710 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,2,2,2,2,
%U A217710 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
%N A217710 Cardinality of the set of possible heights of AVL trees with n (leaf-) nodes.
%C A217710 a(n) increases at Fibonacci numbers (A000045) and decreases at powers of 2 plus 1 (A000051) for n>=8.
%C A217710 a(n) is the height (number of nonzero elements) of column n of triangles A143897, A217298.
%H A217710 Alois P. Heinz, <a href="/A217710/b217710.txt">Table of n, a(n) for n = 1..50000</a>
%H A217710 Wikipedia, <a href="https://en.wikipedia.org/wiki/AVL_tree">AVL tree</a>
%H A217710 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A217710 a(n) = A072649(n) - A029837(n).
%e A217710 a(8) = 2: We have 1 AVL tree with n=8 (leaf-) nodes of height 3 and 16 of height 4 (8 is both Fibonacci number and power of 2):
%e A217710            o              o
%e A217710          /   \          /   \
%e A217710        o       o      o       o
%e A217710       / \     / )    / \     / \
%e A217710      o   o   o  N   o   o   o   o
%e A217710     / ) ( ) ( )    ( ) ( ) ( ) ( )
%e A217710    o  N N N N N    N N N N N N N N
%e A217710   ( )
%e A217710   N N
%p A217710 a:= proc(n) local j, p; for j from ilog[(1+sqrt(5))/2](n)
%p A217710        while combinat[fibonacci](j+1)<=n do od;
%p A217710        p:= ilog2(n);
%p A217710        j-p-`if`(2^p<n, 2, 1)
%p A217710     end:
%p A217710 seq(a(n), n=1..120);
%p A217710 # second Maple program:
%p A217710 a:= proc(n) option remember; `if`(n=0, 1, a(n-1)+
%p A217710      `if`((t-> issqr(t+4) or issqr(t-4))(5*n^2), 1, 0)-
%p A217710      `if`((t-> is(2^ilog2(t)=t))(n-1), 1, 0))
%p A217710     end:
%p A217710 seq(a(n), n=1..120);  # _Alois P. Heinz_, Aug 14 2021
%t A217710 a[n_] := Module[{j, p}, For[j = Log[(1+Sqrt[5])/2, n] // Floor, Fibonacci[j+1] <= n, j++]; p = Log[2, n] // Floor; j-p-If[2^p < n, 2, 1]]; Table[a[n], {n, 1, 120}] (* _Jean-François Alcover_, Dec 30 2013, translated from Maple *)
%Y A217710 Cf. A000045, A000051, A000079, A029837, A072649, A143897, A217298.
%K A217710 nonn
%O A217710 1,8
%A A217710 _Alois P. Heinz_, Mar 20 2013