cp's OEIS Frontend

a(n) is the sum of the absolute values of the (2+n)-th terms in 2^n "random Fibonacci sequences" using either addition or subtraction.

Viswanath's constant V approximates a(15) = 223790 by (2*V)^15 or about 210416.

Approximating a(19) = 7560760 by (2*V)^19 or about 5527978 appears to be bad, why?

Viswanath's constant is not relevant for this sequence, since these two questions are different: what is the growth rate of almost random Fibonacci sequences, what is the average value of the n-th term of such a random Fibonacci sequence? (I've just submitted a paper to Journal of Number Theory to prove that the two problems have different solutions. I'm currently preparing a second paper which gives the explicit value of the constant involved in the context of average value of n-th term.) - Benoit Rittaud (rittaud(AT)math.univ-paris13.fr), Mar 10 2006

Real zero of x^3 + x^2 - x - 2. - Charles R Greathouse IV, May 28 2011

This is the infinite nested radical sqrt(1+sqrt(-1+sqrt(1+sqrt(-1+...)))), evaluated as the limit for an increasing (even) number of terms (an odd number of terms gives always 1) and using the main branch of the complex sqrt(z) function. This real-valued constant is in fact the unique attractor of the complex mapping M(z)=sqrt(1+sqrt(-1+z)), with its attraction domain covering the whole complex plane, excluding z = 1, the other invariant point of M(z). Closely related is A272874. - Stanislav Sykora, May 08 2016

From Wolfdieter Lang, Oct 17 2022: (Start)

This equals r0 - 1/3 where r0 is the real root of y^3 - (4/3)*y - 43/27.

The other roots of x^3 + x^2 - x - 2 are (w1*(4*(43 + 3*sqrt(177)))^(1/3) + w2*(4*(43 - 3*sqrt(177)))^(1/3) - 2)/6 = -1.1027847152... + 0.6654569511...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.

Using hyperbolic functions these roots are -(1 + 2*cosh((1/3)*arccosh(43/16)) - 2*sqrt(3)*sinh((1/3)*arccosh(43/16))*i)/3, and its complex conjugate.

(End)

Suppose b(1) = 1 and b(n+1) = +-b(n) +- x*b(n-1) with the four choices of sign made with equal probability. Embree and Trefethen show that if x is less than this constant, b(n) tends to 0; otherwise, |b(n)| increases without bound. - Charles R Greathouse IV, Jul 19 2013

Named after the American mathematicians Mark Patrick Embree (b. 1974) and Lloyd Nicholas Trefethen (b. 1955). - Amiram Eldar, Jun 16 2021

For each bit of the golden string that is a 1, write seven consecutive bits and for each 0 of the golden string write eight consecutive bits.

Alternate between 0 and 1. Exclude the first seven bits since we are based at zero, just like the golden string. Heads = 1, Tails = 0.

Using the value of Viswanath's constant given by Eric Weisstein's notebook (v = 1.1321506910656020459) makes this sequence different from a(32) forwards; the gap gradually widens after that.

For sufficiently large terms of a random Fibonacci sequence, the powers of Viswanath's constant approximate the absolute value of the terms in such a sequence (with a few notable exceptions).

See Viswanath sequences (A078416 and links) for random sequences of this type.