cp's OEIS Frontend

A220393 A modified Engel expansion of Pi.

%I A220393 #13 Jun 19 2025 03:25:40
%S A220393 1,1,1,8,14,2,2,3,4,5,96,115,8,2,2,2,81,160,2,6,355,140,4,12,6,2,2,3,
%T A220393 4,3,46,66,4,2,9,16,3,4,3,4,2,2,4,9,4,2,4,33,20,2,3,4,2,2
%N A220393 A modified Engel expansion of Pi.
%C A220393 The Engel expansion of a positive real number x is the unique nondecreasing sequence {e(1), e(2), e(3), ...} of positive integers such that x = 1/e(1) + 1/(e(1)*e(2)) + 1/(e(1)*e(2)*e(3)) + .... The terms in the Engel expansion of x are obtained from the iterates of the mapping g(x) := x*(1 + floor(1/x)) - 1 by means of the formula e(n) = 1 + floor(1/g^(n)(x)).
%C A220393 Let h(x) = floor(1/x)*g(x). This is called the harmonic sawtooth map by Crowley. Then the modified Engel expansion of a real number 0 < x <= 1 is a sequence {a(1), a(2), a(3), ...} of positive integers such that x = 1/a(1) + 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) + ... whose terms are obtained from the iterates of the harmonic sawtooth map h(x) by the formulas a(1) = 1 + floor(1/x) and, for n >= 2, a(n) = floor(1/h^(n-2)(x))*{1 + floor(1/h^(n-1)(x))}. Here h^(n)(x) = h(h^(n-1)(x)) denotes the n-th iterate of the map h(x), with the convention that h^(0)(x) = x. For further details see the Bala link.
%C A220393 When the real number x > 1 with, say, floor(x) = m, the modified Engel expansion of x is found by first calculating the modified Engel expansion of x - m and then prepending a sequence of m 1's to this.
%H A220393 Peter Bala, <a href="/A220393/a220393.pdf">A modified Engel expansion</a>
%H A220393 Wikipedia, <a href="http://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>
%F A220393 Let x = Pi - 3. Then a(1) = a(2) = a(3) = 1, a(4) = ceiling(1/x) and, for n >= 1, a(n+4) = floor(1/h^(n-1)(x))*ceiling(1/h^(n)(x)).
%F A220393 Put P(n) = Product_{k = 1..n} a(k). Then we have the Egyptian fraction series expansion Pi = Sum_{n>=1} 1/P(n) = 1/1 + 1/1 + 1/1 + 1/8 + 1/(8*14) + 1/(8*14*2) + 1/(8*14*2*2) + .... For n >= 4, the error made in truncating this series to n terms is less than the n-th term.
%Y A220393 Cf. A006784, A220335, A220336, A220337, A220338, A220394, A220395, A220396, A220397, A220398.
%K A220393 nonn,easy
%O A220393 1,4
%A A220393 _Peter Bala_, Dec 13 2012