This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A269802 #7 Mar 07 2016 12:40:02
%S A269802 6,8,4,7,8,8,2,6,7,5,2,2,6,7,4,7,9,3,3,8,2,4,5,5,8,2,0,0,3,7,0,5,8,3,
%T A269802 3,1,3,2,5,4,7,8,8,5,2,8,6,2,6,3,4,2,3,9,4,6,5,2,8,6,9,2,2,1,6,4,5,1,
%U A269802 2,7,4,6,2,9,8,2,6,9,2,4,1,7,7,8,4,9
%N A269802 Decimal expansion of the number having (1,2,3,4,...) as its Fibonacci-nested interval sequence.
%C A269802 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/F(n+1)) gives the Fibonacci-nested interval sequence of x. Here, F = A000045, the Fibonacci numbers.)
%C A269802 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.
%F A269802 x = sum{1/(P(k)F(k)), k >= 1}, where P(k) = F(1)*F(2)***F(k+2). F = A000045 (Fibonacci numbers).
%e A269802 x = 0.68478826752267479338245582003...
%t A269802 f[n_] := Fibonacci[n]; p[1] = f[3]; p[n_] := p[n - 1] f[n + 2]
%t A269802 Table[p[i]*f[i], {i, 1, 10}]
%t A269802 s = Sum[1/(p[i] f[i]), {i, 1, 200}]; RealDigits[N[s, 100]][[1]]
%Y A269802 Cf. A000027, A000045, A269803.
%K A269802 nonn,cons,easy
%O A269802 0,1
%A A269802 _Clark Kimberling_, Mar 05 2016