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A079585
Decimal expansion of c = (7-sqrt(5))/2.
11
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, 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, 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
OFFSET
1,1
COMMENTS
c is an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Jan 08 2018
From Amiram Eldar, Jul 16 2021: (Start)
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.
Mahler (1975) provided a false proof that this sum is transcendental. The mistake was corrected in Mahler (1976). (End)
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
REFERENCES
Jean-Paul Allouche and Jeffrey Shallit, Automatic sequences, Cambridge University Press, 2003, p. 65.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.2, p. 7.
Ross Honsberger, Mathematical Gems III, Washington, DC: Math. Assoc. Amer., 1985, pp. 135-137.
Alfred S. Posamentier and Ingmar Lehmann, [Phi], The Glorious Golden Ratio, Prometheus Books, 2011, page 75.
LINKS
I. J. Good, A Reciprocal Series of Fibonacci Numbers, Fib. Quart., Vol. 12, No. 4 (1974), p. 346.
History of Science and Mathematics StackExchange, Who was D.A. Millin, the eponym of the Millin Series?, 2022.
Edouard Lucas, Théorie des Fonctions Numériques Simplement Périodiques. [Continued], American Journal of Mathematics, Vol. 1, No. 3 (1878), pp. 197-240. See p. 225, equations 125 and 127.
Kurt Mahler, On the transcendency of the solutions of special class of functional equations, Bull. Austral. Math. Soc., Vol. 13, No. 3 (1975), pp. 389-410.
Kurt Mahler, On the transcendency of the solutions of a special class of functional equations: Corrigendum, Bull. Austral. Math. Soc., Vol. 14, No. 3 (1976), pp. 477-478.
Dale Miller, Publications.
D. A. Millin, Problem H-237, The Fibonacci Quarterly, Vol. 12, No. 3 (1974), p. 309; Sum Reciprocal!, Solution to Problem H-237 by A. G. Shannon, ibid., Vol. 14, No. 2 (1976), pp. 186-187.
Michael Penn, The Millin Series (A nice Fibonacci sum), YouTube video, 2020.
Stanley Rabinowitz, A note on the sum 1/w_{k2^n}, Missouri J. Math. Sci., Vol. 10, No. 3 (1998), pp. 141-146.
Eric Weisstein's World of Mathematics, Millin Series.
FORMULA
c = (7-sqrt(5))/2 = 4 - phi, with phi from A001622.
c = 7/2 - 10*A020837.
c = Sum_{k>=0} 1/F(2^k), where F(k) denotes the k-th Fibonacci number; c = Sum_{k>=0} 1/A058635(k).
Periodic continued fraction representation is [2, 2, 1, 1, 1, 1, ....]. - R. J. Mathar, Mar 24 2011
Minimal polynomial: 11 - 7*x + x^2. - Stefano Spezia, Oct 16 2024
EXAMPLE
c = 2.3819660112501051517954131656343618822796908201942371378645513772947...
MATHEMATICA
RealDigits[4 - GoldenRatio, 10, 111][[1]] (* Robert G. Wilson v, Jan 31 2012 *)
PROG
(PARI) (7 - sqrt(5))/2 \\ Michel Marcus, Sep 05 2017
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, Jan 26 2003
STATUS
approved