A242208 Decimal expansion of log_2(phi), the logarithm to base 2 of phi, the "golden ratio" (1+sqrt(5))/2.

6, 9, 4, 2, 4, 1, 9, 1, 3, 6, 3, 0, 6, 1, 7, 3, 0, 1, 7, 3, 8, 7, 9, 0, 2, 6, 6, 8, 9, 8, 5, 9, 5, 2, 2, 3, 4, 6, 3, 5, 6, 7, 2, 8, 5, 2, 2, 7, 1, 2, 9, 7, 1, 5, 9, 8, 0, 9, 8, 9, 8, 6, 6, 5, 4, 1, 4, 0, 5, 7, 4, 4, 1, 0, 5, 0, 1, 1, 7, 6, 1, 8, 9, 7, 6, 3, 1, 4, 1, 7, 2, 3, 4, 7, 6, 4, 5, 3, 5, 9

Offset: 0

Author

Jean-François Alcover, May 07 2014

Keywords

• nonn
• cons

Comments

The limiting fractal dimension of a pattern generated by cellular automaton rule 150 is 1+log_2(phi).

This number is also involved in the evaluation of asymptotics for the number of odd terms in Pascal's trinomial triangle.

Also, the solution to 1 + 2^x = 4^x. See A328900 for solution to 2^x + 3^x = 4^x. - M. F. Hasler, Oct 30 2019

Examples

			0.6942419136306173017387902668985952234635672852271297159809898665414...
		

Links

• Steven Finch, Pascal Sebah and Zai-Qiao Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008, page 5.
• Daniel Glasscock, Joel Moreira, and Florian K. Richter, Additive transversality of fractal sets in the reals and the integers, arXiv:2007.05480 [math.NT], 2020. See p. 33.
• Stephen Wolfram, Statistical mechanics of cellular automata, page 616.
• Index entries for transcendental numbers.

Crossrefs

Cf. A001622, A328912 (continued fraction).

Programs

Mathematica

RealDigits[Log[2, GoldenRatio], 10, 100] // First

PARI

print(c=log(sqrt(5)+1)/log(2)-1); digits(c\.1^default(realprecision))[^-1] \\ [^-1] to discard possibly incorrect last digit. Use e.g. \p999 to get more digits. - M. F. Hasler, Oct 30 2019

Formula

log((1 + sqrt(5))/2)/log(2).

log(sqrt(5) + 1)/log(2) - 1. - M. F. Hasler, Oct 30 2019