# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a079585 Showing 1-1 of 1 %I A079585 #67 Oct 16 2024 09:21:13 %S A079585 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 A079585 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 A079585 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 A079585 Decimal expansion of c = (7-sqrt(5))/2. %C A079585 c is an integer in the quadratic number field Q(sqrt(5)). - _Wolfdieter Lang_, Jan 08 2018 %C A079585 From _Amiram Eldar_, Jul 16 2021: (Start) %C A079585 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 A079585 Mahler (1975) provided a false proof that this sum is transcendental. The mistake was corrected in Mahler (1976). (End) %C A079585 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 A079585 Jean-Paul Allouche and Jeffrey Shallit, Automatic sequences, Cambridge University Press, 2003, p. 65. %D A079585 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.2, p. 7. %D A079585 Ross Honsberger, Mathematical Gems III, Washington, DC: Math. Assoc. Amer., 1985, pp. 135-137. %D A079585 Alfred S. Posamentier and Ingmar Lehmann, [Phi], The Glorious Golden Ratio, Prometheus Books, 2011, page 75. %H A079585 Chai Wah Wu, Table of n, a(n) for n = 1..10001 %H A079585 I. J. Good, A Reciprocal Series of Fibonacci Numbers, Fib. Quart., Vol. 12, No. 4 (1974), p. 346. %H A079585 History of Science and Mathematics StackExchange, Who was D.A. Millin, the eponym of the Millin Series?, 2022. %H A079585 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. %H A079585 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. %H A079585 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. %H A079585 Dale Miller, Publications. %H A079585 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. %H A079585 Michael Penn, The Millin Series (A nice Fibonacci sum), YouTube video, 2020. %H A079585 Proofwiki, Definition:Millin Series. %H A079585 Stanley Rabinowitz, A note on the sum 1/w_{k2^n}, Missouri J. Math. Sci., Vol. 10, No. 3 (1998), pp. 141-146. %H A079585 Eric Weisstein's World of Mathematics, Millin Series. %H A079585 Index entries for algebraic numbers, degree 2 %F A079585 c = (7-sqrt(5))/2 = 4 - phi, with phi from A001622. %F A079585 c = 7/2 - 10*A020837. %F A079585 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 A079585 Periodic continued fraction representation is [2, 2, 1, 1, 1, 1, ....]. - _R. J. Mathar_, Mar 24 2011 %F A079585 Minimal polynomial: 11 - 7*x + x^2. - _Stefano Spezia_, Oct 16 2024 %e A079585 c = 2.3819660112501051517954131656343618822796908201942371378645513772947... %t A079585 RealDigits[4 - GoldenRatio, 10, 111][[1]] (* _Robert G. Wilson v_, Jan 31 2012 *) %o A079585 (PARI) (7 - sqrt(5))/2 \\ _Michel Marcus_, Sep 05 2017 %Y A079585 Cf. A001622, A020837, A058635. %K A079585 cons,nonn %O A079585 1,1 %A A079585 _Benoit Cloitre_, Jan 26 2003 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE