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A269970 Factorial-nested interval sequence of 1/e. 12
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, 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, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

  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.

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.

Guide to related sequences:

x            factorial-nested interval sequence

1/e              A269970

e-2              A269971

1/pi             A269972

pi-3             A269973

sqrt(1/2)        A269974

sqrt(2)-1        A269975

sqrt(1/3)        A269976

sqrt(3)-1        A269977

1/tau            A269978

A269979          (1,2,3,4,5,6,7,...)

A269980          (1,3,5,7,9,11,...)

A269981          (2,4,6,8,10,13,...)

LINKS

Table of n, a(n) for n=1..86.

CROSSREFS

Cf. A000142, A269971-A269981.

Sequence in context: A280052 A183198 A249160 * A252230 A036043 A128628

Adjacent sequences:  A269967 A269968 A269969 * A269971 A269972 A269973

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Mar 08 2016

STATUS

approved

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Last modified April 4 21:43 EDT 2020. Contains 333238 sequences. (Running on oeis4.)