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A295290 a(n) is the smallest triangular number t such that t - n is a square, or -1 if no such triangular number exists. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

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

FORMULA

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

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

EXAMPLE

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.

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

CROSSREFS

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

Sequence in context: A208539 A174128 A131070 * A165202 A010468 A082009

Adjacent sequences:  A295287 A295288 A295289 * A295291 A295292 A295293

KEYWORD

sign

AUTHOR

Jon E. Schoenfield, Nov 19 2017

STATUS

approved

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Last modified May 25 12:30 EDT 2019. Contains 323568 sequences. (Running on oeis4.)