A269981 Decimal expansion of the number having (2,4,6,8,10,...) = A005843 as its factorial-nested interval sequence.

1, 6, 9, 4, 4, 6, 6, 4, 9, 0, 6, 6, 4, 3, 3, 8, 5, 1, 0, 7, 6, 1, 9, 5, 9, 6, 5, 9, 7, 9, 6, 9, 0, 2, 4, 1, 7, 5, 8, 6, 9, 5, 6, 0, 5, 1, 9, 2, 6, 9, 9, 2, 4, 2, 8, 4, 1, 9, 7, 8, 2, 4, 9, 4, 2, 4, 3, 0, 9, 5, 1, 0, 1, 3, 4, 7, 0, 6, 1, 8, 1, 0, 9, 9, 6, 3

Offset

0,2

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.

See A269970 for a guide to related sequences.

Links

Table of n, a(n) for n=0..85.

Example

x = 0.16944664906643385107619596597969024...

Mathematica

r[n_] := 1/n!; n[k_] := k; Table[n[k], {k, 1, 1000}];
len[1] = r[n[1]] - r[n[1] + 1];
len[k_] := len[k - 1]*(r[n[k]] - r[n[k] + 1])
sum = r[n[1] + 1] + Sum[len[i]*r[n[i + 1] + 1], {i, 1, 300}];
g = N[sum, 150]
RealDigits[g, 10, 100][[1]]

Crossrefs

Cf. A000142, A005843, A269970.

Sequence in context: A143735 A216117 A021148 * A133749 A201130 A248759

Adjacent sequences: A269978 A269979 A269980 * A269982 A269983 A269984

Keyword

nonn,cons,easy

Author

Clark Kimberling, Mar 08 2016

Status

approved