A065806 - OEIS

%I #14 Sep 19 2016 06:07:07

%S 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,

%T 12,14,23,30,17,29,18,15,39,0,25,37,43,0,7,37,6,33,31,40,46,37,8,43,

%U 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

%N A pseudo-random sequence.

%C 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.

%C 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.

%C a(A241671(n)) = 0; a(A241887(n)) = n and a(m) <> n for m < A241887(n). - _Reinhard Zumkeller_, Aug 09 2014

%H Reinhard Zumkeller, <a href="/A065806/b065806.txt">Table of n, a(n) for n = 1..10000</a>

%H Klaus Strassburger, <a href="/A065806/a065806a.jpg">Plot of points [n, A065806(n)], n=1..20000</a>

%H Klaus Strassburger, <a href="/A065806/a065806b.jpg">Plot of points [n,(A065806(n)+1)/(n+3)], n=1..20000</a>

%H <a href="/index/Ps#PRN">Index entries for sequences related to pseudo-random numbers.</a>

%p 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)];

%t 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 *)

%o (Haskell) following the Maple program

%o import Data.IntMap (empty, findWithDefault, insert)

%o a065806 n = a065806_list !! (n-1)

%o a065806_list = f 0 empty where

%o    f i m = (j + 1) : f (i + 1) (insert i (b j) $ insert j bi m) where

%o      j = bi - b (i - 1)

%o      bi = b i

%o      b x = findWithDefault x x m

%o -- _Reinhard Zumkeller_, Aug 09 2014

%Y Cf. A241671, A241887.

%K easy,nice,nonn

%O 1,1

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