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The number of staircase walks on a grid with m horizontal lines and n vertical lines is given by (m+n; m)=((m+n)!)/(m!n!) (Vilenkin 1971, Mohanty 1979, Narayana 1979, Finch ...

A recursive function devised by I. Takeuchi in 1978 (Knuth 1998). For integers x, y, and z, it is defined by (1) This can be described more simply by t(x,y,z)={y if x<=y; {z ...

There are at least two theorems known as Weierstrass's theorem. The first states that the only hypercomplex number systems with commutative multiplication and addition are ...

A totally ordered set (A,<=) is said to be well ordered (or have a well-founded order) iff every nonempty subset of A has a least element (Ciesielski 1997, p. 38; Moore 1982, ...

Let one grain of wheat be placed on the first square of a chessboard, two on the second, four on the third, eight on the fourth, etc. How many grains total are placed on an ...

A zerofree number n is called right truncatable if n and all numbers obtained by successively removing the rightmost digits are prime. There are exactly 83 right truncatable ...

The Beatty sequence is a spectrum sequence with an irrational base. In other words, the Beatty sequence corresponding to an irrational number theta is given by |_theta_|, ...

Consider a Boolean algebra of subsets b(A) generated by a set A, which is the set of subsets of A that can be obtained by means of a finite number of the set operations ...

The golden ratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric ...

In the technical combinatorial sense, an a-ary necklace of length n is a string of n characters, each of a possible types. Rotation is ignored, in the sense that b_1b_2...b_n ...