cp's OEIS Frontend

A269970 Factorial-nested interval sequence of 1/e.

%I A269970 #5 Mar 10 2016 21:01:42
%S A269970 2,1,2,3,1,2,2,1,3,2,1,2,4,1,2,1,2,2,2,1,1,1,2,3,2,4,1,2,2,1,2,2,3,2,
%T A269970 2,2,1,2,1,1,2,1,1,1,1,2,2,2,2,2,3,2,2,2,1,2,1,1,2,1,1,1,1,1,1,1,1,2,
%U A269970 1,1,1,4,3,1,1,1,1,1,1,1,2,2,3,1,1,3
%N A269970 Factorial-nested interval sequence of 1/e.
%C A269970 Suppose that r = (r(n)) is a sequence satisfying (i) 1 = r(1) > r(2) > r(3) > ... and (ii) r(n) -> 0.  For x in (0,1], let n(1) be the index n such that r(n+1) , x <= r(n), and let L(1) = r(n(1))-r(n(1)+1).  Let n(2) be the index n such that r(n(1)+1) < x <= r(n(1)+1) + L(1)r(n), and let L(2) = (r(n(2))-r(r(n)+1)L(1).
%C A269970   Continue inductively to obtain the sequence (n(1), n(2), n(3), ... ), the r-nested interval sequence of x.  Taking r = (1/n!) gives the factorial-nested interval sequence of x.
%C A269970 Conversely, given a sequence s= (n(1),n(2),n(3),...) of positive integers, the number x having satisfying NI(x) = s is the sum of left-endpoints of nested intervals (r(n(k)+1), r(n(k))]; i.e., x = sum{L(k)r(n(k+1)+1), k >=1}, where L(0) = 1.
%C A269970 Guide to related sequences:
%C A269970 x            factorial-nested interval sequence
%C A269970 1/e              A269970
%C A269970 e-2              A269971
%C A269970 1/pi             A269972
%C A269970 pi-3             A269973
%C A269970 sqrt(1/2)        A269974
%C A269970 sqrt(2)-1        A269975
%C A269970 sqrt(1/3)        A269976
%C A269970 sqrt(3)-1        A269977
%C A269970 1/tau            A269978
%C A269970 A269979          (1,2,3,4,5,6,7,...)
%C A269970 A269980          (1,3,5,7,9,11,...)
%C A269970 A269981          (2,4,6,8,10,13,...)
%Y A269970 Cf. A000142, A269971-A269981.
%K A269970 nonn,easy
%O A269970 1,1
%A A269970 _Clark Kimberling_, Mar 08 2016