

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
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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(t1) = 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^21)/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



