A078254 Triangular numbers in which alternate digits are equal.

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 171, 595, 666, 5050, 5151

Offset

1,3

Comments

No other terms found up to A000217(30000). - Michel Marcus, Sep 17 2013

Numbers in which alternate digits are equal have one of two forms: b*(100^n-1)/99 or b*(100^n-1)/99*10+floor(b/10), where 0<=b<=99. These numbers are triangular if 8*b*(100^n-1)/99+1=y^2 and 8*(b*(100^n-1)/99*10+floor(b/10))+1=y^2, respectively. These equations can be reduced to search of integral points on a finite number of the elliptic curves: 8*b*(100^k*x^3-1)/99+1=y^2 and 8*(b*(100^k*x^3-1)/99*10+floor(b/10))+1=y^2, where x=100^floor(n/3) and k=0, 1, or 2. Each of them has a finite number of integral points and hence there is a finite number of terms in this sequence. - Max Alekseyev, Apr 26 2015

I've computed the integral points on the aforementioned elliptic curves and verified that the sequence is complete. - Max Alekseyev, Jul 30 2024

Links

Table of n, a(n) for n=1..19.

Crossrefs

Sequence in context: A069697 A069699 A213516 * A069700 A069698 A069696

Adjacent sequences: A078251 A078252 A078253 * A078255 A078256 A078257

Keyword

nonn,base,fini,full

Author

Amarnath Murthy, Nov 24 2002

Status

approved