A096551 - OEIS

A096551

Consecutive internal states of a linear congruential pseudo-random number generator with a parameter proposed by George Marsaglia as a "candidate for the best of all multipliers".

1, 69069, 475559465, 2801775573, 1790562961, 3104832285, 4238970681, 2135332261, 381957665, 1744831853, 1303896393, 1945705589, 2707602097, 4198202557, 3820321881, 201201733, 2583294017, 4003049741, 2417848425, 1454463253, 3332335313, 2360275549, 2093206905, 2813570789

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OFFSET

1,2

COMMENTS

The period length of 2^30 is only achieved if the starting value a(1) is odd. Even starting values lead to shorter periods, i.e., a starting value that is a multiple of 2^k leads to a sequence with period length 2^(30-k). - Hugo Pfoertner, Nov 21 2024

REFERENCES

D. E. Knuth, The Art of Computer Programming Third Edition. Vol. 2 Seminumerical Algorithms. Chapter 3.3.4 The Spectral Test, Page 108. Addison-Wesley 1997.

G. Marsaglia, The structure of linear congruential sequences, in Applications of Number Theory to Numerical Analysis, (edited by S. K. Zaremba), Academic Press, New York, 249-286, 1972.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

George S. Fishman, Multiplicative Congruential Random Number Generators with Modulus 2^beta: An Exhaustive Analysis for beta = 32 and a Partial Analysis for beta = 48, Math. Comp., 54, 189 (1990), 331-344.

Index entries for sequences related to pseudo-random numbers.

FORMULA

a(1)=1, a(n) = 69069 * a(n-1) mod 2^32. The sequence is periodic with period length 2^30. - corrected by Hugo Pfoertner, Aug 10 2011

a(n) == 1 (mod 4). Hugo Pfoertner, Nov 21 2024

MAPLE

a:= proc(n) option remember; `if`(n<2, n,

irem(69069 *a(n-1), 4294967296))

end:

seq(a(n), n=1..30); # Alois P. Heinz, Jun 10 2014