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By Luke Y.L.

ISBN-10: 0124599400

ISBN-13: 9780124599406

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56] from the French “jointe de culasse de Sierpiński” in [226]. He was led to this inelegant designation (cf. [228, p. 142]) by the second of three methods to obtain this mathematical object, namely starting from a filled (equilateral) triangle to cut out the open middle triangle and repeating this for the 22 Chapter 0. 14). It is remarkable that 800 years before Sierpiński the artists of the Cosmati family in Italy had already anticipated this construction by producing examples of the third repeating step.

The minimum number of colors needed for a proper vertex coloring of a graph G is called its (vertex) chromatic number χ(G). e. the order of the largest complete graph contained in G as a subgraph. e. one that has not yet been assigned to one of its neighbors. This procedure stops after ∣G∣ steps, and the largest number used can not be larger than 1 plus the maximum degree ∆(G) of all vertices of G, because no vertex has more than ∆(G) neighbors. Besides the vertex colorings, there are also edge colorings (cf.

23: A ladder with 4 steps and ∆4 = 10 “pigeons” The question is therefore to determine what for reasons obvious from the figure is called triangular number ∆100 (or ∆4 in the picture). The general term of the sequence ∆ν is defined as the sum of the first ν positive integers or, what amounts to the same, of all natural numbers from 0 to ν ∈ N0 . The solution given in Alcuin’s text, interpreted for general even ν, is to redistribute all entries of the sum into pairs of numbers k and ν − k, k ∈ [ ν2 ]0 , such that only the number ν2 remains unmatched, whence ν ν ∆ν = ⋅ ν + ; 2 2 ] match all entries of the sum, such that for odd ν, the pairs (k, ν − k) for k ∈ [ ν+1 2 0 altogether ν(ν + 1) .

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Algorithms for computations of mathematical functions by Luke Y.L.

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