A178768 Decimal expansion of real constant in an explicit counterexample to the Lagarias-Wang finiteness conjecture.
7, 4, 9, 3, 2, 6, 5, 4, 6, 3, 3, 0, 3, 6, 7, 5, 5, 7, 9, 4, 3, 9, 6, 1, 9, 4, 8, 0, 9, 1, 3, 4, 4, 6, 7, 2, 0, 9, 1, 3, 2, 7, 3, 7, 0, 2, 3, 6, 0, 6, 4, 3, 1, 7, 3, 5, 8, 0, 2, 4, 0, 4, 5, 4, 5, 9, 3, 0, 7, 7, 5, 6, 4, 5, 6, 5, 6, 1, 1, 0, 3, 5, 0, 6, 7, 1, 2
Offset: 0
Examples
0.74932654633036755794396194809134467209132737023606431735802...
Links
- Kevin G. Hare, Ian D. Morris, Nikita Sidorov, and Jacques Theys, An explicit counterexample to the Lagarias-Wang finiteness conjecture, arXiv:1006.2117 [math.OC], 2010-2011.
- Kevin G. Hare, Ian D. Morris, Nikita Sidorov, and Jacques Theys, An explicit counterexample to the Lagarias-Wang finiteness conjecture, Advances in Mathematics 226 (2011), 4667-4701.
- J. C. Lagarias and Y. Wang, The finiteness conjecture for the generalized spectral radius of a set of matrices, Linear Algebra Appl., 214 (1995), 17-42.
Programs
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PARI
t(n) = if (n==0, 1, if (n==1, 2, if (n==2, 2, t(n-1)*t(n-2) - t(n-3)))); \\ A022405 prodinf(n=1, (1 - t(n-1)/(t(n)*t(n+1)))^((-1)^n*fibonacci(n+1))) \\ Michel Marcus, Jun 14 2015; May 10 2019
Formula
Equals Product_{n >= 1} (1 - t(n-1)/(t(n)*t(n+1)))^((-1)^n*Fibonacci(n+1)), where t(n) = A022405(n+1) and Fibonacci(n) = A000045(n). See Theorem 1.1 of Hare et al. (2010, 2011). - Michel Marcus, May 10 2019
Extensions
More terms from Amiram Eldar, May 15 2021