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A079585
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Decimal expansion of c = (7-sqrt(5))/2.
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10
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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
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OFFSET
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1,1
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COMMENTS
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c is an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Jan 08 2018
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
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REFERENCES
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Jean-Paul Allouche and Jeffrey Shallit, Automatic sequences, Cambridge University Press, 2003, p. 65.
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.
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LINKS
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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.
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FORMULA
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c = (7-sqrt(5))/2 = 4 - phi, with phi from A001622.
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
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EXAMPLE
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c = 2.3819660112501051517954131656343618822796908201942371378645513772947...
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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