A079585 Decimal expansion of c = (7-sqrt(5))/2.

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

Author

Benoit Cloitre, Jan 26 2003

Keywords

• cons
• nonn

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

Examples

			c = 2.3819660112501051517954131656343618822796908201942371378645513772947...
		

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

• Chai Wah Wu, Table of n, a(n) for n = 1..10001
• 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.
• Proofwiki, Definition:Millin Series.
• 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.
• Index entries for algebraic numbers, degree 2

Crossrefs

Cf. A001622, A020837, A058635.

Programs

Mathematica

RealDigits[4 - GoldenRatio, 10, 111][[1]] (* Robert G. Wilson v, Jan 31 2012 *)

PARI

(7 - sqrt(5))/2 \\ Michel Marcus, Sep 05 2017

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