A083404 Illustration of Viswanath's constant A078416.

1, 2, 6, 14, 32, 82, 196, 464, 1142, 2746, 6576, 15976, 38484, 92544, 223790, 539402, 1299184, 3136178, 7560760, 18222032, 43956888, 105980632, 255487040, 616137680, 1485562228, 3581617536, 8636505982, 20823634954, 50206996848

Offset: 0

Author

Frank Ellermann, Jun 07 2003

Keywords

• nonn
• easy

Comments

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

Examples

			a(2) = 6 = 1 +1 +3 +abs(-1), the 2^2 last terms in (1,1,0,1), (1,1,0,1), (1,1,2,3), (1,1,2,-1).
		

Links

• B. Rittaud, On the Average Growth of Random Fibonacci Sequences, Journal of Integer Sequences, 10 (2007), Article 07.2.4.

Crossrefs

Cf. Viswanath's constant A078416, V = 1.13198824...

Programs

Rexx

A.1 = 1; B.1 = 1; SSS = 1; do N = 1 to 18; M = 2**(N-1); Sum = 0; do K = 1 to M; L = K + M; ADD = A.K + B.K; SUB = A.K - B.K; A.K = B.K; A.L = B.K; B.K = ADD; B.L = SUB; Sum = Sum + abs( ADD ) + abs( SUB ); end K; SSS = SSS Sum; end N; say SSS

Formula

This sequence is exponentially increasing, with growth rate equal to x-1=1.20556943..., where x is the only real number solution of the equation x^3 = 2x^2 + 1. - Benoit Rittaud (rittaud(AT)math.univ-paris13.fr), Jan 20 2007

Extensions

More terms from David Wasserman, Nov 01 2004