Central polygonal numbers

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Pancakes: With three cuts, there were seven pieces

The central polygonal numbers or, in the English-speaking world, the number sequence of the lazy waiter ( lazy caterer's sequence ) denotes the maximum number of pieces of a cake (discus) that can be achieved with a given number of cuts.

formula

The maximum number of pieces of cake can be created by the specified number of cuts , which must be greater than or equal to zero. ${\displaystyle p}$ ${\displaystyle n}$

${\displaystyle p={\frac {n^{2}+n+2}{2}}}$

This representation is also possible

${\displaystyle p=1+{\binom {n+1}{2}}={\binom {n}{0}}+{\binom {n}{1}}+{\binom {n}{2}}}$.

The result is the following series of numbers, starting with : ${\displaystyle n=0}$

1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, ... (sequence A000124 in OEIS )

By subtracting the number 1, the sequence of central polygonal numbers becomes the sequence of triangular numbers .

proof

The maximum number of pieces, with as few cuts as possible, gives the lazy waiter's row of numbers.

For applies to the number of pieces (whole cake). An (arbitrary) cut ( ) increases the number of pieces by 1 . ${\displaystyle n=0}$ ${\displaystyle p=1}$ ${\displaystyle n=1}$ ${\displaystyle p=2}$

For the -th cut ( ), the maximum number of pieces is achieved by the fact that the new cutting line intersects all previously existing cutting lines inside; the new cutting line must not go through a point of intersection of existing cutting lines. In this way, the -th cut increases the number of pieces . ${\displaystyle n}$ ${\displaystyle n>1}$ ${\displaystyle n-1}$ ${\displaystyle n}$ ${\displaystyle n}$

Overall, this results in the number of pieces

${\displaystyle p=1+\left(1+2+\ldots +(n-1)+n\right)}$.

If you express the sum in brackets using the Gaussian sum formula , you get

${\displaystyle p=1+{\frac {n(n+1)}{2}}={\frac {2+n(n+1)}{2}}={\frac {n^{2}+n+2}{2}}}$,

thereby proving the claim.