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The triangular and square numbers arise naturally as figures one might make in the plane and occurred early in the history of mathematics. They generalize easily to higher dimension and have myriad connections to Pascals triangle, convergents to $\sqrt{2}$ and many other things. There are results with attractive visual proofs. This being so fruitful, it is natural to generalize to pentagonal hexagonal and other planar figurate numbers. The Greeks did so. In part this involves summing gemetric arithmetic progressions, a natural thing to investigate. However (to get to your point) the actual representation as a pattern in the plane is definable but not nearly as nice. (centered polygonal numbers are nicer, but that is another topic).

Let $P(s,n)$ be the $n$th $s$-gonal number. Then $P(s,1)=1$ and , as Gerry notes, one can build up by adding gnomons so $P(s,n+1)=P(s,n)+(1+n(s-2))=\sum_0^n(1+n(s-2)).$ The resulting figures are just not as attractive for $S \gt 4$ as for $s=3$ and $4$. There is not rotational or reflective symmetry of the resulting diagram.

One can complement this picture by looking instead at the transition from $P(s,n)$ to $P(s+1,n).$ Then $P(s+1,n)=P(s,N)+P(3,n-1)=P(3,n)+(s-3)P(3,n-1).$ this corresponds to our view of an $s+1$-gon split into $s-2$ triangles by the diagonals from a selected vertex. If we want them totally disjoint then one triangle is one level larger then the others. This is a generalization of the famous graphical proof for the fact that a square is the sum of two consecutive triangular numbers.

It is reasonable to define $P(2,n)=n$ and then we can also write $P(s+1,n)=P(2,n)+(s-2)P(3,n-1).$

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The triangular and square numbers arise naturally as figures one might make in the plane and occurred early in the history of mathematics. They generalize easily to higher dimension and have myriad connections to Pascals triangle, convergents to $\sqrt{2}$ and many other things. There are results with attractive visual proofs. This being so fruitful, it is natural to generalize to pentagonal hexagonal and other planar figurate numbers. The Greeks did so. In part this involves summing gemetric progressions, a natural thing to investigate. However (to get to your point) the actual representation as a pattern in the plane is definable but not nearly as nice. (centered polygonal numbers are nicer, but that is another topic).

Let $P(s,n)$ be the $n$th $s$-gonal number. Then $P(s,1)=1$ and , as Gerry notes, one can build up by adding gnomons so $P(s,n+1)=P(s,n)+(1+n(s-2))=\sum_0^n(1+n(s-2)).$ The resulting figures are just not as attractive for $S \gt 4$ as for $s=3$ and $4$. There is not rotational or reflective symmetry of the resulting diagram.

One can complement this picture by looking instead at the transition from $P(s,n)$ to $P(s+1,n).$ Then $P(s+1,n)=P(s,N)+P(3,n-1)=P(3,n)+(s-3)P(3,n-1).$ this corresponds to our view of an $s+1$-gon split into $s-2$ triangles by the diagonals from a selected vertex. If we want them totally disjoint then one triangle is one level larger then the others. This is a generalization of the famous graphical proof for the fact that a square is the sum of two consecutive triangular numbers.

It is reasonable to define $P(2,n)=n$ and then we can also write $P(s+1,n)=P(2,n)+(s-2)P(3,n-1).$

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The triangular and square numbers arise naturally as figures one might make in the plane and occurred early in the history of mathematics. They generalize easily to higher dimension and have myriad connections to Pascals triangle, convergents to $\sqrt{2}$ and many other things. There are results with attractive visual proofs. This being so fruitful, it is natural to generalize to pentagonal hexagonal and other planar figurate numbers. The Greeks did so. In part this involves summing gemetric progressions, a natural thing to investigate. However (to get to your point) the actual representation as a pattern in the plane is definable but not nearly as nice. (centered polygonal numbers are nicer, but that is another topic).

Let $P(s,n)$ be the $n$th $s$-gonal number. Then $P(s,1)=1$ and , as Gerry notes, one can build up by adding gnomons so $P(s,n+1)=P(s,n)+(1+n(s-2))=\sum_0^n(1+n(s-2)).$ The resulting figures are just not as attractive for $S \gt 4$ as for $s=3$ and $4$. There is not rotational or reflective symmetry of the resulting diagram.

One can complement this picture by looking instead at the transition from $P(s,n)$ to $P(s+1,n).$ Then $P(s+1,n)=P(s,N)+P(3,n-1)=P(3,n)+(s-3)P(3,n-1).$ this corresponds to our view of an $s+1$-gon split into $s-2$ triangles by the diagonals from a selected vertex. If we want them totally disjoint then one triangle is one level larger then the others. This is a generalization of the famous graphical proof for the fact that a square is the sum of two consecutive triangular numbers.

It is reasonable to define $P(2,n)=n$ and then we can also write $P(s+1,n)=P(2,n)+(s-2)P(3,n-1).$