A308085 a(n) is the least positive number k such that n*(n-1)/2 + k*(k-1)/2 is a square.

1, 1, 2, 3, 4, 2, 6, 7, 1, 9, 10, 6, 3, 13, 14, 2, 16, 17, 18, 4, 6, 21, 3, 23, 24, 9, 5, 27, 13, 4, 30, 31, 2, 6, 34, 35, 5, 37, 38, 16, 7, 30, 23, 6, 44, 20, 46, 8, 16, 1, 7, 51, 12, 53, 9, 42, 23, 8, 58, 59, 60, 10, 27, 63, 9, 65, 20, 67, 11, 69, 6, 10, 72, 3, 44, 12, 21, 77, 11, 34, 80, 46

Offset: 1

Author

J. M. Bergot and Robert Israel, Jun 05 2019

Keywords

• nonn
• look

Comments

a(n) <= n-1 if n > 1, because (n-1)*(n-2)/2 + n*(n-1)/2 = (n-1)^2.

a(7*k+2) <= k and a(7*k+6) <= k+2, because (7*k+2)*(7*k+1)/2 + k*(k-1)/2 = (5*k+1)^2 and (7*k+6)*(7*k+5)/2 + (k+2)*(k+1)/2 = (5*k+4)^2.

From Bernard Schott, Jun 27 2019: (Start)

a(m) = 1 iff the triangular number t(m-1) = (m-1)*m/2 is a square, so iff m-1 is in A001108, or m in A055997.

a(m) = 2 iff the triangular number t(m-1) + 1 is a square, so iff m-1 is in A006451. (End)

Examples

			a(5) = 4 because 4*3/2 + 5*4/2 = 4^2 and none of 1*0/2 + 5*4/2, 2*1/2 + 5*4/2, 3*2/2 + 5*4/2 are squares.
		

Links

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

Crossrefs

Cf. A000217, A055997 (a(n)=1).

Programs

Maple

f:= proc(n) local k;
  for k from 1 do
    if issqr((k*(k-1)+n*(n-1))/2) then return k fi
  od
end proc:
map(f, [$1..100]);

PARI

a(n) = {my(k=1); while (!issquare(n*(n-1)/2 + k*(k-1)/2), k++); k;} \\ Michel Marcus, Jun 27 2019