A180928 - OEIS

%I #23 Feb 17 2020 14:56:36

%S 0,1,5,27

%N 1 + product of any two terms is a triangular number.

%C The sequence could also start 0, 1, 2, 27, ... - _R. J. Mathar_, Oct 03 2010

%C This sequence is also finite and complete. - _Max Alekseyev_, Feb 16 2011

%C If 1*x+1, 5*x+1, 27*x+1 are triangular numbers, then 8*x+9=p^2, 40*x+9=q^2, 216*x+9=r^2 for some integers p,q,r. They should also satisfy the system of equations { 5*p^2 - q^2 = 36, 27*p^2 - r^2 = 234 } which has no integer solutions. See Alekseyev, 2011.

%C A192225 contains another result from the same paper.

%H Max A. Alekseyev (2011). <a href="http://www.integers-ejcnt.org/vol11a.html">On the Intersections of Fibonacci, Pell, and Lucas Numbers</a>, INTEGERS 11(3), pp. 239-259. doi:<a href="http://dx.doi.org/10.1515/INTEG.2011.021">10.1515/INTEG.2011.021</a>

%e (0*1)+1 = 1 is triangular.

%e (0*5)+1 = 1 is triangular.

%e (1*5)+1 = 6 is triangular.

%e (0*27)+1 = 1 is triangular.

%e (1*27)+1 = 28 is triangular.

%e (5*27)+1 = 136 is triangular.

%e (0*70)+1 = 1 is triangular.

%e (1*70)+1 = 71 is NOT triangular, so 70 is not the next value.

%e (5*70)+1 = 351 is triangular.

%e (27*70)+1 = 1891 is triangular.

%Y This is to A030063 as A000217 is to A000290.

%K nonn,full,fini

%O 1,3

%A _Jonathan Vos Post_, Sep 25 2010

%E No further terms below 10^20. - _Charles R Greathouse IV_, Sep 29 2010

%E Keywords 'full', 'fini' from _Max Alekseyev_, Feb 16 2011