1911 Encyclopædia Britannica/Polygonal Numbers

POLYGONAL NUMBERS, in mathematics. Suppose we have a number of equal circular counters, then the number of counters which can be placed on a regular polygon so that the tangents to the outer rows form the regular polygon and all the internal counters are in contact with its neighbours, is a “polygonal number ” of the order of the polygon. If the polygon be a triangle then it is readily seen that the numbers are 3, 6, IO, 1 5 and generally en (n + 1); if a square, 4, 9, 16, . and generally n2; if a pentagon, 5, 12, 22 and generally n(3n—1); if a. hexagon, 6, 15, 28, . and generally n(2n- 1); and similarly for a polygon of r sides, the general expression for the corresponding polygonal number is § n[(n- 1) (V-2)+2]

Algebraically, polygonal numbers may be regarded as the sums of consecutive terms of the arithmetical progressions having 1 for the first term and I, 2, 3, . . for the common differences. Taking unit common difference we have the series 1; 1+2=3; 1+2+3 =6; 1+2-+-3-1-4=1o; or generally I-l-2-f-3 -}- n§ n(n+I); these are triangular numbers. With a common difference 2 we have I: l+3=4; I+3+5=9; I +3+5+7= 16? or generally I-l-3-l-5+ . -l- (2n-1)=n2; and generally for the polygonal number of the rth order we take the sums of consecutive terms of the series

1, 1+(r-2), 1+2 (r-2), . 1+n-1 r-2;

and hence the nth polygonal number of the rth order is the sum of n terms of this series, i, e.,

x+1+(r-2)+1+2(r-2)+ -1-(1-}-n-1.r-2)

=n+§ n.n-1.r-2.

The series I, 2, 3, 4, ... or generally n, are the so-called"' linear numbers " (cf. FIGURATE NUMBERS).