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Sequence of the Day for December 1

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.





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For more details about today's Sequence of the Day, see {{Sequence of the Day for December 1}}.

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