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Wonders of numbers. Adventures in mathematics, mind, and meaning. (English) Zbl 0983.00008

Oxford: Oxford University Press. xx, 396 p. (2001).
The lecture of this book can be compared with a visit of an art gallery. In the first part the visitor finds some attractive “pen and paper” puzzles, occasionally with surprising solutions. Then we enter the cafeteria and can listen to some math gossip. Next is the main part of the gallery, showing a nearly overwhelming number of mostly abstract pictures. We find some well known titles such as prime numbers, Fibonacci numbers, Catalan numbers, perfect numbers and of course amicable numbers, but there are many more. But the “pen and paper approach” is not of much use here anymore. At least a notebook is needed with a software that can handle recursive formulas. The visitor will then be highly motivated to take a break and create such a picture of its own. The last part brings him back to the world of puzzles and games. Also here, many of the beauties can only be gained with the help of a computer, but it is highly comforting that some simple algebra occasionally do just as well.

MSC:

00A08 Recreational mathematics
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Online Encyclopedia of Integer Sequences:

Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).
Decimal expansion of square root of 5.
a(n) = 2^(n-1)*(2^n - 1), n >= 0.
Decimal expansion of sqrt(2*Pi).
a(n) = 10^n*(10^n+1)/2.
A ternary tribonacci triangle: form the triangle as follows: start with 3 single values: 1, 2, 3. Each succeeding row is a concatenation of the previous 3 rows.
”Madonna’s Sequence”: add 1 (mod 10) to each digit of Pi.
Form triangle as follows: start with three single digits: 0, 1, 2. Each succeeding row is a concatenation of the previous three rows.
Decimal expansion of Pi^e.
a(n) = binomial(n+1, 2)^5.
Decimal expansion of 6/Pi^2.
Ulam numbers starting with the numbers 1 and 9.
Schizophrenic sequence: these are the repeating digits in the decimal expansion of sqrt(f(2n+1)), where f(m) = A014824(m).
Katydid sequence: closed under n -> 2n + 2 and 7n + 7.
Fixed point of the morphism 1 -> 121, 2 -> 122, starting from 1.
Images of centered hexamorphic numbers: suppose k-th centered hexagonal number H_c(k) (A003215) ends in k; sequence gives value of H_c(k).
Square pyramorphic numbers: suppose the k-th square pyramidal number S(k) (A000330) ends in k; sequence gives value of S(k).
a(1) = 1; a(2) = 2; a(3) = 3; a(n) is concatenation of a(n-3), a(n-2) and a(n-1).
a(1) = 1; a(2) = 2; a(3) = 3; a(n) is concatenation of a(n-3), a(n-2) and a(n-1); the digits on the right of the first 3 (if any) are then swapped with the digits on the left.
Fixed point of the morphism 1 -> 121, 2 -> 12, starting from 1.
Numbers that are not differences between successive Ulam numbers (A002858).
Next n digits of e, base of the natural logarithms.
Total number of zeros in the decimal expansions of 2^n and 5^n.
Square pyramorphic numbers: integers m such that the sum of the first m squares (A000330) ends in m.
Numbers n such that the last 9 decimal digits of the n-th Fibonacci number is pandigital 1-9.
Numbers for which the smallest number of steps to reach 1 in ”3x+1” (or Collatz) problem is a prime.
Primes in A171490.