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

1,5

Comments

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

Links

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

Example

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

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
    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 *)

Crossrefs

A309507 counts all triples with k>0.

Cf. A000217, A068194.

Sequence in context: A219644 A193676 A029291 * A369363 A022872 A091423

Adjacent sequences: A333526 A333527 A333528 * A333530 A333531 A333532

Keyword

nonn,look

Author

N. J. A. Sloane, Mar 31 2020

Status

approved