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A083404 Illustration of Viswanath's constant A078416. 0

%I #15 Nov 03 2019 13:15:12

%S 1,2,6,14,32,82,196,464,1142,2746,6576,15976,38484,92544,223790,

%T 539402,1299184,3136178,7560760,18222032,43956888,105980632,255487040,

%U 616137680,1485562228,3581617536,8636505982,20823634954,50206996848

%N Illustration of Viswanath's constant A078416.

%C 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.

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

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

%C 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

%H B. Rittaud, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Rittaud2/rittaud11.html">On the Average Growth of Random Fibonacci Sequences</a>, Journal of Integer Sequences, 10 (2007), Article 07.2.4.

%F 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

%e 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).

%o (Rexx)

%o 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

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

%K nonn,easy

%O 0,2

%A _Frank Ellermann_, Jun 07 2003

%E More terms from _David Wasserman_, Nov 01 2004

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Last modified March 29 06:15 EDT 2024. Contains 371265 sequences. (Running on oeis4.)