A220398 A modified Engel expansion of the golden ratio (1/2)*(1 + sqrt(5)) (A001622).

1, 2, 5, 8, 3, 4, 4, 6, 2, 162, 322, 2, 51842, 103682, 2, 5374978562, 10749957122, 2, 57780789062419261442, 115561578124838522882, 2, 6677239169351578707225356193679818792962, 13354478338703157414450712387359637585922, 2

Offset

1,2

Comments

See A220393 for a definition of the modified Engel expansion of a positive real number. For further details see the Bala link.

Links

Table of n, a(n) for n=1..24.

Peter Bala, A modified Engel expansion

S. Crowley, Integral transforms of the harmonic sawtooth map, the Riemann zeta function, fractal strings, and a finite reflection formula, arXiv:1210.5652 [math.NT], 2012-2020

Wikipedia, Engel Expansion

Formula

Let h(x) = x*(floor(1/x) + (floor(1/x))^2) - floor(1/x). Let x = 1/2*(1 + sqrt(5)) - 1. Then a(1) = 1, a(2) = ceiling(1/x) and, for n >= 1, a(n+2) = floor(1/h^(n-1)(x))*(1 + floor(1/h^(n)(x))).

Recurrence equations: For n >= 3, a(3*n) = 2. For n >= 4 we have a(3*n+2) = 2*a(3*n+1) - 2 and a(3*n+1) = 2*(a(3*n-2) - 1)^2.

Put P(n) = Product_{k = 1..n} a(k). Then we have the Egyptian fraction series expansion sqrt(2) = Sum_{n>=1} 1/P(n) = 1 + 1/2 + 1/(2*5) + 1/(2*5*8) + 1/(2*5*8*3) + 1/(2*5*8*3*4) + .... For n >= 2, the error made in truncating this series to n terms is less than the n-th term.

Crossrefs

Cf. A001622, A028259, A220335, A220336, A220337, A220338, A220393, A220394, A220395, A220396, A220397.

Sequence in context: A220337 A198545 A296430 * A352633 A200225 A258749

Adjacent sequences: A220395 A220396 A220397 * A220399 A220400 A220401

Keyword

nonn,easy

Author

Peter Bala, Dec 13 2012

Status

approved