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Let p(d) be the probability that a random walk on a d-D lattice returns to the origin. In 1921, Pólya proved that p(1)=p(2)=1, (1) but p(d)<1 (2) for d>2. Watson (1939), ...

Pronic numbers are figurate numbers of the form P_n=2T_n=n(n+1), where T_n is the nth triangular number. The first few are 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, ... (OEIS ...

A sequence produced by the instructions "reverse, add to the original, then sort the digits." For example, after 668, the next iteration is given by 668+866=1534, so the next ...

In 1913, Ramanujan asked if the Diophantine equation of second order 2^n-7=x^2, sometimes called the Ramanujan-Nagell equation, has any solutions other than n=3, 4, 5, 7, and ...

The regular hexagon is the regular polygon with six sides, as illustrated above. The inradius r, circumradius R, sagitta s, and area A of a regular hexagon can be computed ...

Consider a game, first proposed by Nicolaus Bernoulli, in which a player bets on how many tosses of a coin will be needed before it first turns up heads. The player pays a ...

A series-reduced tree is a tree in which all nodes have degree other than 2 (in other words, no node merely allows a single edge to "pass through"). Series-reduced trees are ...

Ramanujan calculated mu=1.45136380... (Hardy 1999, Le Lionnais 1983, Berndt 1994), while the correct value is mu=1.45136923488... (OEIS A070769; Derbyshire 2004, p. 114). The ...

There are many unsolved problems in mathematics. Several famous problems which have recently been solved include: 1. The Pólya conjecture (disproven by Haselgrove 1958, ...

The Somos sequences are a set of related symmetrical recurrence relations which, surprisingly, always give integers. The Somos sequence of order k, or Somos-k sequence, is ...