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%I #60 Feb 29 2024 01:52:55
%S 2,3,8,1,9,6,6,0,1,1,2,5,0,1,0,5,1,5,1,7,9,5,4,1,3,1,6,5,6,3,4,3,6,1,
%T 8,8,2,2,7,9,6,9,0,8,2,0,1,9,4,2,3,7,1,3,7,8,6,4,5,5,1,3,7,7,2,9,4,7,
%U 3,9,5,3,7,1,8,1,0,9,7,5,5,0,2,9,2,7,9,2,7,9,5,8,1,0,6,0,8,8,6,2,5,1,5,2,4
%N Decimal expansion of c = (7-sqrt(5))/2.
%C c is an integer in the quadratic number field Q(sqrt(5)). - _Wolfdieter Lang_, Jan 08 2018
%C From _Amiram Eldar_, Jul 16 2021: (Start)
%C Sum_{k>=0} 1/F(2^k) is sometimes called "Millin series" after D. A. Millin, a high school student at Annville, Pennsylvania, who posed in 1974 the problem of proving that it equals (7-sqrt(5))/2. This identity was in fact already known to Lucas in 1878.
%C Mahler (1975) provided a false proof that this sum is transcendental. The mistake was corrected in Mahler (1976). (End)
%C The name "Millin" was a misprint of "Miller", the author of the problem was Dale A. Miller. His name was corrected in the solution to the problem (1976). - _Amiram Eldar_, Feb 29 2024
%D Jean-Paul Allouche and Jeffrey Shallit, Automatic sequences, Cambridge University Press, 2003, p. 65.
%D Ross Honsberger, Mathematical Gems III, Washington, DC: Math. Assoc. Amer., 1985, pp. 135-137.
%D Alfred S. Posamentier and Ingmar Lehmann, [Phi], The Glorious Golden Ratio, Prometheus Books, 2011, page 75.
%H Chai Wah Wu, <a href="/A079585/b079585.txt">Table of n, a(n) for n = 1..10001</a>
%H I. J. Good, <a href="http://www.fq.math.ca/Scanned/12-4/good.pdf">A Reciprocal Series of Fibonacci Numbers</a>, Fib. Quart., Vol. 12, No. 4 (1974), p. 346.
%H History of Science and Mathematics StackExchange, <a href="https://hsm.stackexchange.com/questions/14434/who-was-d-a-millin-the-eponym-of-the-millin-series">Who was D.A. Millin, the eponym of the Millin Series?</a>, 2022.
%H Edouard Lucas, <a href="https://www.jstor.org/stable/2369311">Théorie des Fonctions Numériques Simplement Périodiques. [Continued]</a>, American Journal of Mathematics, Vol. 1, No. 3 (1878), pp. 197-240. See p. 225, equations 125 and 127.
%H Kurt Mahler, <a href="https://doi.org/10.1017/S0004972700024643">On the transcendency of the solutions of special class of functional equations</a>, Bull. Austral. Math. Soc., Vol. 13, No. 3 (1975), pp. 389-410.
%H Kurt Mahler, <a href="https://doi.org/10.1017/S0004972700025430">On the transcendency of the solutions of a special class of functional equations: Corrigendum</a>, Bull. Austral. Math. Soc., Vol. 14, No. 3 (1976), pp. 477-478.
%H Dale Miller, <a href="https://www.lix.polytechnique.fr/Labo/Dale.Miller/papers/pubs.html">Publications</a>.
%H D. A. Millin, <a href="https://www.fq.math.ca/Scanned/12-3/advanced12-3.pdf">Problem H-237</a>, The Fibonacci Quarterly, Vol. 12, No. 3 (1974), p. 309; <a href="https://www.fq.math.ca/Scanned/14-2/advanced14-2.pdf">Sum Reciprocal!</a>, Solution to Problem H-237 by A. G. Shannon, ibid., Vol. 14, No. 2 (1976), pp. 186-187.
%H Michael Penn, <a href="https://www.youtube.com/watch?v=oVUr1G9pSD4">The Millin Series (A nice Fibonacci sum)</a>, YouTube video, 2020.
%H Proofwiki, <a href="https://proofwiki.org/wiki/Definition:Millin_Series">Definition:Millin Series</a>.
%H Stanley Rabinowitz, <a href="https://doi.org/10.35834/1998/1003141">A note on the sum 1/w_{k2^n}</a>, Missouri J. Math. Sci., Vol. 10, No. 3 (1998), pp. 141-146.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MillinSeries.html">Millin Series</a>.
%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F c = (7-sqrt(5))/2 = 4 - phi, with phi from A001622.
%F c = 7/2 - 10*A020837.
%F c = Sum_{k>=0} 1/F(2^k), where F(k) denotes the k-th Fibonacci number; c = Sum_{k>=0} 1/A058635(k).
%F Periodic continued fraction representation is [2, 2, 1, 1, 1, 1, ....]. - _R. J. Mathar_, Mar 24 2011
%e c = 2.3819660112501051517954131656343618822796908201942371378645513772947...
%t RealDigits[4 - GoldenRatio, 10, 111][[1]] (* _Robert G. Wilson v_, Jan 31 2012 *)
%o (PARI) (7 - sqrt(5))/2 \\ _Michel Marcus_, Sep 05 2017
%Y Cf. A001622, A020837, A058635.
%K cons,nonn
%O 1,1
%A _Benoit Cloitre_, Jan 26 2003
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