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, 10, 1 5 ... and generally z n (n -{- r); if a square, 4, 9, 16, ... and generally n'; if a pentagon, 5, 12, 22 ... and generally n(3n--I); if a hexagon, 6, 1 5, 28, ... and generally n(2nI); and similarly for a polygon of r sides, the general expression for the corresponding polygonal number is 2n[(n-I) ( r- 2)+2].
Algebraically, polygonal numbers may be regarded as the sums of consecutive terms of. the arithmetical progressions having I for the first term and 1, 2, 3, ... for the common differences. Taking unit common difference we have the series I; I +2 =3; 1+2+3 = 6; I + 2 + 3 + 4 = to; or generally I + 2 + 3 ... + n = Zn(n+I); these are triangular numbers. With a common difference 2 we have I; 1 +3 = 4; 1 +3+5 = 9; I+3+5+7=16; or generally 1+3+5+ ... -i- (2n-I)=n 2; and generally for the polygonal number of the rth order we take the sums of consecutive terms of the series 1, I+(r-2), 1+2 (r-2), ... 1+n-I.r-2; and hence the nth polygonal number of the rth order is the sum of n terms of this series, i.e., I+I +(r2)+I +2(r-2)+. .. -}-(I Fn-I.r-2) =n+2n.n-1.r The series I, 2, 3, 4, ... or generally n, are the so-called "linear numbers" (Cf. Figurate Numbers).