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Decimal expansion of a limit associated with the asymptotic number of ways of writing a number as a sum of powers of 2, with each power used at most twice (cardinality of "alternating bit sets" of a given number, also known as Stern's diatomic sequence).
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%I #21 May 08 2020 16:13:35

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

%T 9,5,7,4,8,6,8,2,9,9,1,0,0,5,8,5,6,9,1,4,9,1,0,0,1,9,6,7,9,2,5,9,6,5,

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

%N Decimal expansion of a limit associated with the asymptotic number of ways of writing a number as a sum of powers of 2, with each power used at most twice (cardinality of "alternating bit sets" of a given number, also known as Stern's diatomic sequence).

%C See A002487.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.16.3 Alternating bit sets, p. 148.

%H Neil J. Calkin and Herbert S. Wilf, <a href="http://www.math.clemson.edu/~calkin/Papers/test-bak.pdf">Binary Partitions of Integers and Stern-Brocot-Like Trees</a>, 1998. Section 10, open question 9 (which is answered by Coons and Tyler).

%H Michael Coons and Jason Tyler, <a href="http://arxiv.org/abs/1307.1521">The maximal order of Stern's diatomic sequence.</a> arXiv:1307.1521 [math.NT], 2013-2014.

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020. See p. 20.

%F 3^(log(phi)/log(2))/sqrt(5), where phi is the golden ratio.

%e 0.95885419082476738320909430420365929574868299100585691491...

%t RealDigits[ 3^(Log[GoldenRatio]/Log[2]) / Sqrt[5], 10, 103] // First

%Y Cf. A002487.

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Sep 03 2014