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

0,1

Links

Table of n, a(n) for n=0..104.

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).

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)

Example

0.429947433342452720114697035519922323324065011589304617040276...
= log(1.53717671718235794959014032895522160250150809343236...)

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

Prog

(PARI) (1/4)*suminf(k=1, (log((1/3)*(2^(k+2) - (-1)^k))/2^k)) \\ Michel Marcus, May 14 2020

Crossrefs

Cf. A338294.

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.

Sequence in context: A144811 A185654 A228041 * A179398 A233295 A298567

Adjacent sequences: A242046 A242047 A242048 * A242050 A242051 A242052

Keyword

nonn,cons

Author

Jean-François Alcover, Aug 13 2014

Status

approved