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### Square Number

A square number, also called a perfect square, is a figurate number of the form , where is an integer. The square numbers for , 1, ... are 0, 1, 4, 9, 16, 25, 36, 49, ... (OEIS A000290).A plot of the first few square numbers represented as a sequence of binary bits is shown above. The top portion shows to , and the bottom shows the next 510 values.The generating function giving the squarenumbers is(1)The st square number is given in terms of the th square number by(2)since(3)which is equivalent to adding a gnomon to the previoussquare, as illustrated above.The th square number is equal to the sum of the st and th triangular numbers,(4)(5)as can seen in the above diagram, in which the st triangular number is represented by the white triangles, the th triangular number is represented by the black triangles, and the total number of triangles is the square number (R. Sobel, pers. comm.).Square numbers can also be generated by taking the product of two consecutive..

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