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A220337 A modified Engel expansion for 3*sqrt(15) - 11. 9
2, 5, 8, 2, 32, 62, 2, 1922, 3842, 2, 7380482, 14760962, 2, 108942999582722, 217885999165442, 2, 23737154316161495960243527682, 47474308632322991920487055362, 2, 1126904990058528673830897031906808442930637286502826475522 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

For a brief description of the modified Engel expansion of a real number see A220335.

Let p >= 2 be an integer and set Q(p) = (p - 1)*sqrt(p^2 - 1) - (p^2 - p - 1), so Q(4) = 3*sqrt(15) - 11. Iterating the identity Q(p) = 1/2 + 1/(2*(p+1)) + 1/(2*(p+1)*(2*p)) + 1/(2*(p+1)*(2*p))*Q(2*p^2-1) leads to a representation for Q(p) as an infinite series of unit fractions. The sequence of denominators of these unit fractions can be used to find the modified Engel expansion of Q(p). For further details see the Bala link. The present sequence is the case p = 4. For other cases see A220335 (p = 2), A220336 (p = 3) and A220338 (p = 5).

LINKS

Table of n, a(n) for n=1..20.

P. Bala, A modified Engel expansion for certain quadratic irrationals

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]

Wikipedia, Engel Expansion

FORMULA

Define the harmonic sawtooth map h(x) := floor(1/x)*(x*ceiling(1/x) - 1). Let x = 3*sqrt(15) - 11. Then a(1) = ceiling(1/x) and for n >= 2, a(n) = floor(1/h^(n-2)(x))*ceiling(1/h^(n-1)(x)), where h^(n)(x) denotes the n-th iterate of the map h(x), with the convention h^(0)(x) = x.

a(3*n+2) = 1/2*{2 + (4 + sqrt(15))^(2^n) + (4 - sqrt(15))^(2^n)} and

a(3*n+3) = (4 + sqrt(15))^(2^n) + (4 - sqrt(15))^(2^n), both for n >= 0.

For n >= 0, a(3*n+1) = 2. For n >= 1, a(3*n+2) = 2*(A005828(n-1))^2 and a(3*n+3) = 4*(A005828(n-1))^2 - 2.

Recurrence equations:

For n >= 1, a(3*n+2) = 2*{a(3*n-1)^2 - 2*a(3*n-1) + 1} and

a(3*n+3) = 2*a(3*n+2) - 2.

Put P(n) = product(k = 0..n} a(k). Then we have the infinite Egyptian fraction representation 3*sqrt(15) - 11 = sum {n >=0} 1/P(n) = 1/2 + 1/(2*5) + 1/(2*5*8) + 1/(2*5*8*2) + 1/(2*5*8*2*32) + ....

CROSSREFS

Cf. A005828, A220335 (p = 2), A220336 (p = 3), A220338 (p = 5).

Sequence in context: A248510 A020772 A131598 * A198545 A296430 A220398

Adjacent sequences:  A220334 A220335 A220336 * A220338 A220339 A220340

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Dec 12 2012

STATUS

approved

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Last modified February 22 04:58 EST 2020. Contains 332115 sequences. (Running on oeis4.)