A239970 Least positive k such that triangular(k) + triangular(n+k) is a triangular number (A000217).

2, 5, 3, 6, 9, 4, 15, 18, 21, 5, 27, 30, 9, 36, 6, 42, 10, 48, 51, 14, 7, 60, 63, 15, 69, 72, 19, 8, 81, 26, 20, 13, 17, 24, 99, 9, 105, 14, 111, 114, 29, 120, 123, 126, 10, 132, 135, 34, 20, 144, 147, 35, 153, 45, 11, 29, 165, 33, 17, 174, 30, 44, 183, 186, 189, 12, 18, 23

Offset

0,1

Comments

a(n) <= 3*n-3, because triangular(3*n-3) + triangular(4*n-3) = triangular(5*n-4).

In other words, smallest k>0 such that 8*k^2 + 4*(2*k + 1)*n + 4*n^2 + 8*k + 1 = m^2 has an integer solution. - Ralf Stephan, Apr 01 2014

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Example

n=5: the smallest solution k>0 to 8*k^2 + 64*k + 225 = m^2 is k=4, so a(5)=4.

Prog

(PARI) triangular(n) = n*(n+1)/2;
is_triangular(n) = issquare(8*n+1);
s=[]; for(n=0, 100, k=1; while(!is_triangular(triangular(k)+triangular(n+k)), k++); s=concat(s, k)); s \\ Colin Barker, Mar 31 2014
(PARI) a(n)=my(k=1); while(!ispolygonal(k*(k+n+1)+(n^2+n)/2, 3), k++); k \\ Charles R Greathouse IV, Apr 01 2014
(Haskell)
a239970 n = head [k | k <- [1..],
                      a010054 (a000217 k + a000217 (n + k)) == 1]
-- Reinhard Zumkeller, Apr 03 2014

Crossrefs

Cf. A000217, A239969.

Cf. A010054.

Sequence in context: A335499 A358838 A359008 * A111202 A194280 A163362

Adjacent sequences: A239967 A239968 A239969 * A239971 A239972 A239973

Keyword

nonn

Author

Alex Ratushnyak, Mar 30 2014

Extensions

First PROG corrected by Colin Barker, Apr 04 2014

Status

approved