cp's OEIS Frontend

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.

%I A242049 #26 Feb 14 2021 13:22:29
%S A242049 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,
%T A242049 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,
%U A242049 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
%N 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.
%H A242049 Steven Finch, Pascal Sebah and Zai-Qiao Bai, <a href="http://arXiv.org/abs/0802.2654">Odd Entries in Pascal's Trinomial Triangle</a>, arXiv:0802.2654 [math.NT], 2008, p. 14.
%H A242049 Sara Kropf and Stephan Wagner, <a href="https://arxiv.org/abs/1605.03654">q-Quasiadditive functions</a>, arXiv:1605.03654 [math.CO], 2016.  See section 5 example 8 mean mu for the case s_n is the Jacobsthal sequence.
%H A242049 Kevin Ryde, <a href="http://user42.tuxfamily.org/pari-vpar/index.html">vpar</a> examples/complete-binary-matchings.gp calculations and code in PARI/GP, see log(C).
%F A242049 Equals (1/4)*Sum_{k >= 1} (log((1/3)*(2^(k+2) - (-1)^k))/2^k).
%F A242049 From _Kevin Ryde_, Feb 13 2021: (Start)
%F A242049 Equals log(A338294).
%F A242049 Equals Sum_{k>=1} (1/k)*( 1/(1+(-2)^(k+1)) - 1/(-3)^k ) (an alternating series).
%F A242049 (End)
%e A242049 0.429947433342452720114697035519922323324065011589304617040276...
%e A242049 = log(1.53717671718235794959014032895522160250150809343236...)
%t A242049 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
%o A242049 (PARI) (1/4)*suminf(k=1, (log((1/3)*(2^(k+2) - (-1)^k))/2^k)) \\ _Michel Marcus_, May 14 2020
%Y A242049 Cf. A338294.
%Y A242049 Cf. A242208 (1+x+x^2)^n, A242021 (1+x+x^3)^n, A242022 (1+x+x^2+x^3+x^4)^n, A241002 (1+x+x^4)^n, A242047 (1+x+...+x^4+x^5)^n, A242048 (1+x+...+x^5+x^6)^n.
%K A242049 nonn,cons
%O A242049 0,1
%A A242049 _Jean-François Alcover_, Aug 13 2014