A338305 -id:A338305 - cp's OEIS frontend

A192222 a(n) = Fibonacci(2^n + 1).

1, 2, 5, 34, 1597, 3524578, 17167680177565, 407305795904080553832073954, 229265413057075367692743352179590077832064383222590237

Offset: 0

Jonathan Sondow, Jun 26 2011

a(n) is the numerator of the n-th iterate when Newton's method is applied to the function x^2 - x - 1 with initial guess x = 1. The n-th iterate is a(n)/A058635(n). - Jason Zimba, Jan 20 2023

John Gill and Matthew Miller, Newton's Method and Ratios of Fibonacci Numbers, Fibonacci Quarterly, 19(1):1-3, February 1981.
Jonathan Sondow, Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers, Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, Vol. 1385, No. 1 (2011), pp. 97-100, arXiv preprint, arXiv:1106.4246 [math.NT], 2011.
Yohei Tachiya, Transcendence of certain infinite products, J. Number Theory, Vol. 125, No. 1 (2007), pp. 182-200.

Cf. A000045 (Fibonacci numbers F(n)), A001622, A134973 (decimal expansion of 3/phi), A192223 (Lucas(2^n + 1)), A338305.

Table[Fibonacci[2^n + 1], {n, 0, 10}] (* T. D. Noe, Jan 11 2012 *)

a(n) = A000045(2^n + 1).
Product_{n>0} (1 + 1/a(n)) = 3/phi = A134973, where phi = (1+sqrt(5))/2 is the golden mean.
Sum_{n>=0} 1/a(n) = A338305. - Amiram Eldar, Oct 22 2020

A338304 Decimal expansion of Sum_{k>=0} 1/L(2^k), where L(k) is the k-th Lucas number (A000032).

Original entry on oeis.org

1, 4, 9, 7, 9, 2, 0, 3, 8, 0, 9, 9, 9, 0, 6, 2, 7, 1, 9, 8, 7, 0, 6, 8, 5, 5, 5, 3, 9, 9, 2, 8, 5, 9, 6, 0, 8, 0, 7, 2, 0, 7, 7, 1, 9, 8, 5, 7, 0, 8, 5, 9, 7, 0, 4, 0, 4, 9, 3, 2, 2, 3, 9, 8, 9, 5, 4, 0, 5, 2, 7, 7, 6, 9, 5, 3, 2, 2, 3, 7, 8, 3, 9, 9, 3, 2, 1

Offset: 1

Amiram Eldar, Oct 21 2020

Erdős and Graham (1980) asked whether this constant is irrational or transcendental.
Badea (1987) proved that it is irrational, and André-Jeannin (1991) proved that it is not a quadratic irrational in Q(sqrt(5)), in contrast to the corresponding sum with Fibonacci numbers, Sum_{k>=0} 1/F(2^k) = (7-sqrt(5))/2 (A079585).
Bundschuh and Pethö (1987) proved that it is transcendental.

			1.49792038099906271987068555399285960807207719857085...

Richard André-Jeannin, A note on the irrationality of certain Lucas infinite series, The Fibonacci Quarterly, Vol. 29, No. 2 (1991), pp. 132-136.
Catalin Badea, The irrationality of certain infinite series, Glasgow Mathematical Journal, Vol. 29, No. 2 (1987), pp. 221-228.
Peter Bundschuh and Attila Pethö, Zur transzendenz gewisser Reihen, Monatshefte für Mathematik, Vol. 104, No. 3 (1987), pp. 199-223, alternative link.
Paul Erdős and Ronald L. Graham, Old and new problems and results in combinatorial number theory, L'enseignement Mathématique, Université de Genève, 1980, pp. 64-65.
Index entries for transcendental numbers

Cf. A000045, A000032, A001566, A079585, A338305.

RealDigits[Sum[1/LucasL[2^n], {n, 0, 10}], 10, 100][[1]]

Equals 1 + Sum_{k>=0} 1/A001566(k).

cp's OEIS Frontend

A192222 a(n) = Fibonacci(2^n + 1).

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