A269979 Decimal expansion of the number having (1,2,3,4,...) = A000027 as its factorial-nested interval sequence.

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

Offset: 0

Author

Clark Kimberling, Mar 08 2016

Keywords

• nonn
• cons
• easy

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.

Examples

			x = 0.59045235435205548168124328101350...
		

Crossrefs

Cf. A000027, A000142, A269970.

Programs

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]]