A220337 A modified Engel expansion for 3*sqrt(15) - 11.

2, 5, 8, 2, 32, 62, 2, 1922, 3842, 2, 7380482, 14760962, 2, 108942999582722, 217885999165442, 2, 23737154316161495960243527682, 47474308632322991920487055362, 2, 1126904990058528673830897031906808442930637286502826475522

Offset: 1

Author

Peter Bala, Dec 12 2012

Keywords

• nonn
• easy

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

• Peter Bala, A modified Engel expansion for certain quadratic irrationals
• Wikipedia, Engel Expansion

Crossrefs

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

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