A309228 a(n) is the greatest possible height of a binary tree where all nodes hold positive squares and all interior nodes also equal the sum of their two children and the root node has value n^2.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 3, 2, 1, 1, 1, 3, 2, 1, 3, 1, 3, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 3, 3, 1, 2, 1, 1, 3, 1, 2, 3, 1, 1, 1, 4, 1, 1, 3, 1, 2, 1, 1, 3, 3, 3, 1, 1, 3, 1, 2, 1, 3, 1, 1, 4, 1, 3

Offset: 1

Author

Rémy Sigrist, Jul 16 2019

Keywords

• nonn

Comments

The sequence is unbounded and for any k > 0, A309167(k) is the least n such that a(n) = k.

Examples

			a(1) = 1:
              1^2
               |
a(5) = 2:
           3^2    4^2
            \     /
             \   /
              5^2
               |
a(13) = 3:
          3^2    4^2
           \     /
            \   /
             5^2    12^2
              \      /
               \    /
                13^2
                  |
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A004144, A309167.

Programs

Maple

f:= proc(n) option remember; local S,x,y;
   S:= map(t -> subs(t,[x,y]), {isolve(x^2+y^2=n^2)});
   S:= select(t -> type(t,list(posint)) and t[2]>=t[1], S);
   if S = {} then 1 else 1+max(map(procname,map(op,S))) fi
end proc:
map(f, [$1..100]); # Robert Israel, Feb 27 2022

Mathematica

a = Table[1, {m = 100}];
Do[Do[If[IntegerQ@ Sqrt[v2 = n^2-u^2], a[[n]] = Max[a[[n]], 1+Max[a[[u]], a[[Floor@ Sqrt[v2]]]]]], {u, 1, n-1}], {n, 1, m}];
Table[a[[n]], {n, 1, m}] (* Jean-François Alcover, Aug 19 2022, after PARI code *)

PARI

a = vector(87,n,1); for (n=1, #a, for (u=1, n-1, if (issquare(v2=n^2-u^2), a[n]=max(a[n],1+max(a[u],a[sqrtint(v2)])))); print1 (a[n]", "))

Formula

a(n) = 1 iff n belongs to A004144.

a(A309167(n)) = n.

If n^2 = u^2 + v^2 with u > v > 0, then a(n) >= 1 + max(a(u), a(v)).