Pyramid number

In mathematics , pyramidal numbers or pyramidal numbers are a class of polyhedral numbers , that is, three-dimensional figured numbers . Some authors use the term pyramidal number for the special case of quadratic pyramidal numbers. They are the three-dimensional generalizations of the plane polygonal numbers .

calculation

The -th -angular pyramidal number can be expressed with the formula ${\displaystyle n}$ ${\displaystyle k}$

${\displaystyle {\frac {n(n+1)[(k-2)n-k+5]}{6}}}$

to calculate.

Alternatively, the -th -angular pyramidal number can be calculated as the sum of the first -angular polygonal numbers . ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle k}$

Pyramid numbers for polygons with few corners

${\displaystyle k}$	designation	Explicit formula	the first values	Generating function
3	Tetrahedral numbers	${\displaystyle {\tfrac {1}{6}}n(n+1)(n+2)}$	(0,) 1, 4, 10, 20, 35,… (sequence A000292 in OEIS )	${\displaystyle {\frac {x}{(x-1)^{4}}}}$
4th	Square pyramidal numbers	${\displaystyle {\tfrac {1}{6}}n(n+1)(2n+1)}$	(0,) 1, 5, 14, 30, 55,… (sequence A000330 in OEIS )	${\displaystyle {\frac {x(x+1)}{(x-1)^{4}}}}$
5	Pentagonal pyramidal numbers	${\displaystyle {\tfrac {1}{2}}n^{2}(n+1)}$	(0,) 1, 6, 18, 40, 75,… (sequence A002411 in OEIS )	${\displaystyle {\frac {x(2x+1)}{(x-1)^{4}}}}$
6th	Hexagonal pyramidal numbers	${\displaystyle {\tfrac {1}{6}}n(n+1)(4n-1)}$	(0,) 1, 7, 22, 50, 95,… (sequence A002412 in OEIS )	${\displaystyle {\frac {x(3x+1)}{(x-1)^{4}}}}$
7th	Heptagonal pyramidal numbers	${\displaystyle {\tfrac {1}{6}}n(n+1)(5n-2)}$	(0,) 1, 8, 26, 60, 115,… (sequence A002413 in OEIS )	${\displaystyle {\frac {x(4x+1)}{(x-1)^{4}}}}$

Note: Some authors include the zero as the zeroth or first figured number, others do not.

Further connections with other figured numbers

The -th quadratic pyramidal number can also be derived from the -th triangular number and the -th tetrahedral number according to the formula ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \Delta _{n}}$ ${\displaystyle (n-1)}$ ${\displaystyle T_{n-1}}$

${\displaystyle \Delta _{n}+2\cdot T_{n-1}}$

to calculate.

In addition, the quadratic pyramidal numbers are the sum of two consecutive tetrahedral numbers.

The sum of the first tetrahedral numbers results in a pentatope number , a four-dimensional figured number .

Individual evidence

1. ↑ Eric W. Weisstein : Pyramidal Numbers . In: MathWorld (English).
2. ↑ Eric W. Weisstein : Square Pyramidal Numbers . In: MathWorld (English).
3. ↑ Eric W. Weisstein : Pentagonal Pyramidal Numbers . In: MathWorld (English).
4. ↑ Eric W. Weisstein : Hexagonal Pyramidal Numbers . In: MathWorld (English).
5. ↑ Eric W. Weisstein : Heptagonal Pyramidal Numbers . In: MathWorld (English).