About: Square pyramidal number   Sponge Permalink

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that is, by adding up the squares of the first n integers, or it was suggested that "by multiplying the nth pronic number by the nth odd number" you got the required result. But on further observation this is not true. By mathematical induction it is possible to derive one formula from the other. An equivalent formula is given in Fibonacci's Liber Abaci (1202, ch. II.12). This is a special case of Faulhaber's formula. The first few pyramid numbers are: 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819 (sequence A000330 in OEIS).

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  • Square pyramidal number
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  • that is, by adding up the squares of the first n integers, or it was suggested that "by multiplying the nth pronic number by the nth odd number" you got the required result. But on further observation this is not true. By mathematical induction it is possible to derive one formula from the other. An equivalent formula is given in Fibonacci's Liber Abaci (1202, ch. II.12). This is a special case of Faulhaber's formula. The first few pyramid numbers are: 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819 (sequence A000330 in OEIS).
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  • SquarePyramidalNumber
Title
  • Square Pyramidal Number
abstract
  • that is, by adding up the squares of the first n integers, or it was suggested that "by multiplying the nth pronic number by the nth odd number" you got the required result. But on further observation this is not true. By mathematical induction it is possible to derive one formula from the other. An equivalent formula is given in Fibonacci's Liber Abaci (1202, ch. II.12). This is a special case of Faulhaber's formula. The first few pyramid numbers are: 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819 (sequence A000330 in OEIS). Pyramid numbers can be modelled in physical space with a given number of balls and a square frame that hold in place the number of balls forming the base, that is, n2. They also solve the problem of counting the number of squares in an n × n grid.
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