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## Sequence of the Day for December 1

A188892: Numbers $\scriptstyle n \,$ such that there is no triangular $\scriptstyle n \,$-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 $\scriptstyle n \,$-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 $\scriptstyle x^2 + x \,=\, (n - 2)y^2 - (n - 4)y \,$ has no integer solutions $\scriptstyle x \,\geq\, y \,>\, 1 \,$, as conversion to a generalized Pell equation shows that if $\scriptstyle n \,=\, k^2 + 2 \,$, then the first equation has only a finite number of solutions. From there one can pinpoint those values of $\scriptstyle n \,$ that produce no integer solutions greater than 1.