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There are several fractal curves associated with Sierpiński. The area for the first Sierpiński curve illustrated above (Sierpiński curve 1912) is A=1/3(7-4sqrt(2)). The curve ...

The Sierpiński sieve is a fractal described by Sierpiński in 1915 and appearing in Italian art from the 13th century (Wolfram 2002, p. 43). It is also called the Sierpiński ...

The Sierpiński carpet is the fractal illustrated above which may be constructed analogously to the Sierpiński sieve, but using squares instead of triangles. It can be ...

The nth-order Sierpiński carpet graph is the connectivity graph of black squares in the nth iteration of the Sierpiński carpet fractal. The first three iterations are shown ...

The Sierpiński gasket graph of order n is the graph obtained from the connectivity of the Sierpiński sieve. The first few Sierpiński gasket graphs are illustrated above. S_2 ...

The nth-order Sierpiński tetrahedron graph is the connectivity graph of black triangles in the nth iteration of the tetrix fractal. The first three iterations are shown ...

Let the sum of squares function r_k(n) denote the number of representations of n by k squares, then the summatory function of r_2(k)/k has the asymptotic expansion ...

As proved by Sierpiński (1960), there exist infinitely many positive odd numbers k such that k·2^n+1 is composite for every n>=1. Numbers k with this property are called ...

A Sierpiński number of the first kind is a number of the form S_n=n^n+1. The first few are 2, 5, 28, 257, 3126, 46657, 823544, 16777217, ... (OEIS A014566). Sierpiński proved ...

A Sierpiński number of the second kind is a number k satisfying Sierpiński's composite number theorem, i.e., a Proth number k such that k·2^n+1 is composite for every n>=1. ...