A065806 A pseudo-random sequence.

2, 0, 4, 0, 5, 2, 8, 0, 7, 3, 8, 10, 10, 3, 6, 17, 0, 12, 16, 4, 21, 2, 18, 21, 5, 22, 22, 3, 16, 12, 14, 23, 30, 17, 29, 18, 15, 39, 0, 25, 37, 43, 0, 7, 37, 6, 33, 31, 40, 46, 37, 8, 43, 17, 22, 30, 25, 19, 43, 8, 10, 50, 49, 60, 14, 36, 45, 28, 56, 38, 33, 26, 68, 61, 63, 72, 72, 3

Offset

1,1

Comments

The sequence looks like a realization of a sequence of independent random variables x(n), with probability function P(x(n) = i) = 1/(n+2), i=0..n+1. The sequence (a(n)+1)/(n+3) seems to be a good U(0,1) pseudo-random number generator.

Conjectures: -1 < a(n) < n+2; If a(n) = n+1 then a(n+1) = 0; All integers >1 occur in the sequence; For each c in [0,1] there exists a subsequence a(i_j) with (a(i_j)+1)/(i_j+3)-> c, j->infinity.

a(A241671(n)) = 0; a(A241887(n)) = n and a(m) <> n for m < A241887(n). - Reinhard Zumkeller, Aug 09 2014

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Klaus Strassburger, Plot of points [n, A065806(n)], n=1..20000

Klaus Strassburger, Plot of points [n,(A065806(n)+1)/(n+3)], n=1..20000

Index entries for sequences related to pseudo-random numbers.

Maple

N := 2000: b := array(-N..N, [seq(i, i=-N..N)]): s := 0; for i from 0 to N do; j := (b[i]-b[i-1]); s := s+1; a[s] := j+1; if j > N or j < -N then break; end if; c := b[i]; b[i] := b[j]; b[j] := c; end do: a := [seq(a[l], l=1..s)];

Mathematica

n = 2000; Clear[b]; b[i_] := b[i] = i; s = 0; For[i = 0, i <= n, i++, j = b[i] - b[i-1]; s++; a[s] = j+1; If[j > n || j < -n, Break[]]; c = b[i]; b[i] = b[j]; b[j] = c]; Table[a[l], {l, 1, s}] (* Jean-François Alcover, Sep 19 2016, adapted from Maple *)

Prog

(Haskell) following the Maple program
import Data.IntMap (empty, findWithDefault, insert)
a065806 n = a065806_list !! (n-1)
a065806_list = f 0 empty where
   f i m = (j + 1) : f (i + 1) (insert i (b j) $ insert j bi m) where
     j = bi - b (i - 1)
     bi = b i
     b x = findWithDefault x x m
-- Reinhard Zumkeller, Aug 09 2014

Crossrefs

Cf. A241671, A241887.

Sequence in context: A035647 A357487 A225437 * A331185 A228087 A320582

Adjacent sequences: A065803 A065804 A065805 * A065807 A065808 A065809

Keyword

easy,nice,nonn

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Nov 21 2001

Status

approved