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Schröder numbers

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1: 26.6 Other Lattice Path Numbers
Schröder Number r ( n )
26.6.4 r ( n ) = D ( n , n ) - D ( n + 1 , n - 1 ) , n 1 .
Table 26.6.4: Schröder numbers r ( n ) .
n r ( n ) n r ( n ) n r ( n ) n r ( n ) n r ( n )
2: Bibliography S
  • M. R. Schroeder (2006) Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity. 4th edition, Springer-Verlag, Berlin.
  • 3: 27.17 Other Applications
    §27.17 Other Applications
    Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. … There are also applications of number theory in many diverse areas, including physics, biology, chemistry, communications, and art. Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.
    4: 27.16 Cryptography
    §27.16 Cryptography
    The primes are kept secret but their product n = p q , an 800-digit number, is made public. …With the most efficient computer techniques devised to date (2010), factoring an 800-digit number may require billions of years on a single computer. For this reason, the codes are considered unbreakable, at least with the current state of knowledge on factoring large numbers. … For further information see Apostol and Niven (1994, p. 24), and for other applications to cryptography see Menezes et al. (1997) and Schroeder (2006).