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Template:Sequence of the Day for December 1

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Intended for: December 1, 2011

Timetable

  • First draft entered by Alonso del Arte on August 28, 2011
  • Draft reviewed by Daniel Forgues on December 1, 2011
  • Draft to be approved by November 1, 2011
Yesterday's SOTD * Tomorrow's SOTD

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A188892: Numbers
n
such that there is no triangular
n
-gonal number greater than 1.
{ 11, 18, 38, 102, 198, 326, 486, ... }
The triangular numbers are essentially the building blocks of the other figurate numbers. Therefore it is rather surprising that there can be sequences of
n
-gonal numbers that don’t overlap with the sequence of triangular numbers at all (other than 0 and 1). T. D. Noe has demonstrated that the equation
x 2 + x = (n  −  2) y 2  −  (n  −  4) y
has no integer solutions
x   ≥   y > 1
, as conversion to a generalized Pell equation shows that if
n = k  2 + 2
, then the first equation has only a finite number of solutions. From there one can pinpoint those values of
n
that produce no integer solutions greater than 1.