A242049 Decimal expansion of 'lambda', the Lyapunov exponent characterizing the asymptotic growth rate of the number of odd coefficients in Pascal trinomial triangle mod 2, where coefficients are from (1+x+x^2)^n.
4, 2, 9, 9, 4, 7, 4, 3, 3, 3, 4, 2, 4, 5, 2, 7, 2, 0, 1, 1, 4, 6, 9, 7, 0, 3, 5, 5, 1, 9, 9, 2, 2, 3, 2, 3, 3, 2, 4, 0, 6, 5, 0, 1, 1, 5, 8, 9, 3, 0, 4, 6, 1, 7, 0, 4, 0, 2, 7, 6, 0, 7, 2, 5, 7, 4, 2, 8, 3, 3, 7, 2, 8, 3, 1, 3, 9, 8, 1, 0, 5, 6, 8, 4, 5, 6, 3, 4, 9, 0, 0, 7, 4, 8, 4, 7, 4, 2, 5, 3, 6, 6, 5, 4, 3
Offset: 0
Examples
0.429947433342452720114697035519922323324065011589304617040276... = log(1.53717671718235794959014032895522160250150809343236...)
Links
- Steven Finch, Pascal Sebah and Zai-Qiao Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008, p. 14.
- Sara Kropf and Stephan Wagner, q-Quasiadditive functions, arXiv:1605.03654 [math.CO], 2016. See section 5 example 8 mean mu for the case s_n is the Jacobsthal sequence.
- Kevin Ryde, vpar examples/complete-binary-matchings.gp calculations and code in PARI/GP, see log(C).
Crossrefs
Programs
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Mathematica
digits = 105; lambda = (1/4)*NSum[Log[(1/3)*(2^(k+2) - (-1)^k)]/2^k, {k, 1, Infinity}, WorkingPrecision -> digits + 5, NSumTerms -> 500]; RealDigits[lambda, 10, digits] // First -
PARI
(1/4)*suminf(k=1, (log((1/3)*(2^(k+2) - (-1)^k))/2^k)) \\ Michel Marcus, May 14 2020
Formula
Equals (1/4)*Sum_{k >= 1} (log((1/3)*(2^(k+2) - (-1)^k))/2^k).
From Kevin Ryde, Feb 13 2021: (Start)
Equals log(A338294).
Equals Sum_{k>=1} (1/k)*( 1/(1+(-2)^(k+1)) - 1/(-3)^k ) (an alternating series).
(End)