A295290 a(n) is the smallest triangular number t such that t - n is a square, or -1 if no such triangular number exists.

0, 1, 3, 3, -1, 6, 6, -1, -1, 10, 10, 15, 21, -1, 15, 15, -1, 21, -1, 28, 21, 21, -1, -1, 28, -1, -1, 28, 28, 45, 55, -1, 36, -1, -1, 36, 36, -1, -1, 55, -1, 45, 78, -1, 45, 45, 55, -1, -1, -1, 66, 55, -1, 78, 55, 55, 105, 66, -1, -1, -1, -1, 66, -1, -1, 66

Offset: 0

Author

Jon E. Schoenfield, Nov 19 2017

Keywords

• sign

Comments

Smallest triangular number (A000217) that exceeds a square by exactly n, or -1 if there is no such triangular number.

Examples

			a(0) = 0 because 0 is the smallest number that is both triangular and square.
a(12) = 21 because 21 - 12 = 9 = 3^2 and there is no triangular number t < 21 such that t - 12 is a square.
a(4) = -1 because there exists no triangular number t such that t - 4 is a square.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000217 (triangular numbers), A000290 (squares), A001100 (square triangular numbers).

Programs

Maple

f:= proc(n) local s,t,R, v, R0;
  R:= [isolve(s^2 - 2*t^2 = 8*n+1)];
  if R = [] then return -1 fi;
  v:= indets(R,name) minus {s,t};
  R0:= remove(hastype,eval(R,v[1]=0),negative);
  s:= subs(R0[1],s);
  (s^2-1)/8;
end proc:
map(f, [$0..100]); # Robert Israel, Nov 22 2017

Mathematica

a[n_] := Module[{s, t, k}, If[Solve[s^2 - 2t^2 == 8n+1, {s, t}, Integers] == {}, Return[-1]]; For[k = 0, True, k++, t = k(k+1)/2; If[IntegerQ[ Sqrt[t-n]], Return[t]]]];
a /@ Range[0, 100] (* Jean-François Alcover, Oct 16 2020 *)

Formula

a(t) = t for every triangular number t.

a(t-1) = t for every positive triangular number t.