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The maximum number of pieces from consecutive cuts are the numbers in the Lazy Caterer's Sequence. When a circle is cut n times to produce the maximum number of pieces, represented as p = f (n), the n th cut must be considered; the number of pieces before the last cut is f (n − 1), while the number of pieces added by the last cut is n.
The maximum number of pieces, p obtainable with n straight cuts is the n-th triangular number plus one, forming the lazy caterer's sequence (OEIS A000124) One way of calculating the depreciation of an asset is the sum-of-years' digits method, which involves finding T n, where n is the length in years of the asset's useful life.
The cake numbers are the 3-dimensional analogue of the 2-dimensional lazy caterer's sequence. The difference between successive cake numbers also gives the lazy caterer's sequence. [1] Cake numbers (blue) and other OEIS sequences in Bernoulli's triangle. The fourth column of Bernoulli's triangle (k = 3) gives the cake numbers for n cuts, where ...
As the third column of Bernoulli's triangle (k = 2) is a triangular number plus one, it forms the lazy caterer's sequence for n cuts, where n ≥ 2. [4] The fourth column (k = 3) is the three-dimensional analogue, known as the cake numbers, for n cuts, where n ≥ 3. [5]
The numbers along the left edge of the triangle are the lazy caterer's sequence and the numbers along the right edge are the triangular numbers. The nth row sums to n(n 2 + 1)/2, the constant of an n × n magic square (sequence A006003 in the OEIS).
The number of points (n), chords (c) and regions (r G) for first 6 terms of Moser's circle problem. In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes called Moser's circle problem (named after Leo Moser), has a solution by an inductive method.
Sequences of weight distribution codes often omit periodically recurring zeros. For example, consider: the prime numbers, the palindromic primes, the Fibonacci sequence, the lazy caterer's sequence, and the coefficients in the series expansion of (+) (). In OEIS lexicographic order, they are:
The number of iterations needed for , to reach a fixed point is the Kaprekar function's persistence of , and undefined if it never reaches a fixed point. There are only a finite number of p {\displaystyle p} -Kaprekar numbers and cycles for a given base b {\displaystyle b} , because if n = b p + m {\displaystyle n=b^{p}+m} , where m > 0 ...