A333529 Number of triples [n,k,m] with n <= k <= m satisfying T_n + T_k = T_m, where T_i = i*(i+1)/2 are the triangular numbers.

0, 1, 1, 1, 2, 2, 1, 2, 3, 2, 3, 3, 2, 5, 3, 1, 4, 4, 3, 5, 5, 2, 3, 4, 4, 5, 5, 3, 6, 6, 1, 3, 5, 6, 7, 5, 2, 5, 6, 3, 6, 6, 3, 8, 9, 2, 3, 4, 6, 8, 6, 3, 6, 11, 5, 6, 5, 2, 7, 7, 2, 9, 5, 3, 11, 6, 3, 6, 11, 6, 5, 5, 2, 9, 9, 6, 11, 6, 3, 7, 7, 2, 7, 12, 6, 5, 7, 3, 10, 16, 6, 6, 5, 6, 6, 3, 4, 12, 12, 5, 6, 6, 3, 12

Offset: 1

Author

N. J. A. Sloane, Mar 31 2020

Keywords

• nonn
• look

Comments

a(n)=1 if n is in A068194. - Robert Israel, Apr 03 2020

Examples

			There is a list of all triples (including those with 0 < k < n) with n <= 16 in A309507.
		

Links

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

Crossrefs

A309507 counts all triples with k>0.

Cf. A000217, A068194.

Programs

Maple

with(numtheory):
A:=[]; M:=150; ct:=Array(0..M,0):
for n from 1 to M do
TT:=n*(n+1);
dlis:=divisors(TT);
  for d in dlis do
if (d mod 2) = 1 then e := TT/d;
mi:=min(d,e); ma:=max(d,e);
k:=(ma-mi-1)/2; m:=(ma+mi-1)/2;
# skip if k=n then
    ct[n]:=ct[n]+1;
    lprint(n,k,m);
    fi;
fi;
od:
od:
[seq(ct[n],n=1..M)];
# alternative:
f:= proc(n) local t,t0, r, dmax, divs;
    t:= n*(n+1);
    r:= padic:-ordp(t,2);
    t0:= t/2^r;
    dmax:= floor((sqrt(8*t+1)-1)/2-n);
    divs:= numtheory:-divisors(t0);
    nops(select(`<=`,divs,dmax)) + nops(select(`<=`,divs,dmax/2^r))
end proc:
map(f, [$1..200]); # Robert Israel, Apr 03 2020

Mathematica

T[n_] := n(n+1)/2;
r[n_] := Reduce[n <= k <= m && T[n] + T[k] == T[m], {k, m}, Integers];
a[n_] := Module[{rn = r[n], r0}, r0 = rn[[0]]; Which[r0 === Or, Length[rn], r0 === And, 1, rn === False, 0, True, Print["error ", n, " ", rn]]];
Array[a, 100] (* Jean-François Alcover, Jun 08 2020 *)