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In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. Square pyramidal numbers also solve the problem of counting the number of squares in an grid. ==Formula== The first few square pyramidal numbers are: : , ... . These numbers can be expressed in a formula as : This is a special case of Faulhaber's formula, and may be proved by a mathematical induction.〔Hopcroft, Motwani & Ullman (2007), (p. 20 )〕 An equivalent formula is given in Fibonacci's Liber Abaci (1202, ch. II.12). In modern mathematics, figurate numbers are formalized by the Ehrhart polynomials. The Ehrhart polynomial of a polyhedron is a polynomial that counts the number of integer points in a copy of that is expanded by multiplying all its coordinates by the number . The Ehrhart polynomial of a pyramid whose base is a unit square with integer coordinates, and whose apex is an integer point at height one above the base plane, is .〔.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「square pyramidal number」の詳細全文を読む スポンサード リンク
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