A096555 Consecutive internal states of the linear congruential pseudo-random number generator RANDU that was used in the IBM Scientific Subroutine Library for IBM System/360 computers in the 1970's.

1, 65539, 393225, 1769499, 7077969, 26542323, 95552217, 334432395, 1146624417, 1722371299, 14608041, 1766175739, 1875647473, 1800754131, 366148473, 1022489195, 692115265, 1392739779, 2127401289, 229749723, 1559239569, 845238963, 1775695897, 899541067, 153401569

Offset: 1

Author

Hugo Pfoertner, Jul 19 2004

Keywords

• nonn
• easy

Comments

Due to a poor choice of the multiplier the generator fails most 3-d criteria for randomness. 9*a(n-2)-6*a(n-1)+a(n) = 0 mod 2^31. This was first described by George Marsaglia. The animated gif in the link demonstrates the deficient behavior. The animation shows 10000 points, each of whose coordinate triples (x,y,z) were formed from successive outputs of the generator. From a suitable view angle, it can be seen that these points do not fill the 3-D space, but lie in a few planes parallel to each other.

References

• D. E. Knuth, The Art of Computer Programming Third Edition. Vol. 2 Seminumerical Algorithms. Chapter 3.3.4 The Spectral Test, Page 107. Addison-Wesley 1997.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Sarah Belet, 'Round the Twist, Blog Entry, Friday May 16 2014
• George Marsaglia, Random numbers fall mainly in the planes, Proceedings of the National Academy of Sciences, 61, 25-28, 1968.
• Hugo Pfoertner, Animation showing the deficient 3-d behavior, 2024.
• Index entries for sequences related to pseudo-random numbers.
• Index entries for linear recurrences with constant coefficients, order 536870912.

Crossrefs

Cf. A096550-A096561 for other pseudo-random number generators.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, n,
      irem(65539 *a(n-1), 2147483648))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 10 2014

Mathematica

NestList[Mod[#*65539, 2^31] &, 1, 50] (* Paolo Xausa, Aug 29 2024 *)

PARI

a(n)=lift(Mod(65539,2^31)^(n-1)) \\ Charles R Greathouse IV, Jan 13 2016

Formula

a(1)=1, a(n) = 65539*a(n-1) mod 2^31. The sequence is periodic with period length 2^29.