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A105394
Decimal expansion of sum of reciprocals of squares of Lucas numbers.
4
1, 2, 0, 7, 2, 9, 1, 9, 9, 6, 9, 8, 5, 7, 4, 7, 0, 7, 4, 4, 1, 7, 2, 0, 4, 1, 8, 4, 2, 5, 7, 6, 9, 9, 9, 4, 5, 3, 0, 6, 9, 2, 1, 4, 5, 4, 0, 1, 9, 0, 3, 6, 3, 7, 6, 9, 5, 1, 3, 1, 1, 5, 9, 4, 2, 2, 1, 2, 2, 4, 0, 0, 1, 5, 4, 0, 7, 0, 3, 5, 7, 7, 6, 1, 6, 7, 7, 6, 5, 5, 9, 7, 8, 6, 8, 8, 9, 9, 9, 2
OFFSET
1,2
COMMENTS
This constant is transcendental (Duverney et al., 1997). - Amiram Eldar, Oct 30 2020
REFERENCES
Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Wiley, 1987, p. 97.
LINKS
Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, C. R. Acad. Sci. Paris Ser. I Math., Vol. 308, No. 19 (1989), pp. 539-541.
Paul S. Bruckman,, Problem H-347, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 20, No. (1982), p. 372; It All Adds Up, Solution to Problem H-347 by the proposer, ibid., Vol. 22, No. 1 (1984), pp. 94-96.
Daniel Duverney, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa, Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 7 (1997), pp. 140-142.
Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant.
FORMULA
Equals Sum_{n >= 1} 1/L(n)^2.
Equals (1/8)*( theta_3(beta)^4 - 1 ), where beta = (3 - sqrt(5))/2 and theta_3(q) = 1 + 2*Sum_{n >= 1} q^(n^2) is a theta function. See Borwein and Borwein, Exercise 7(f), p. 97. - Peter Bala, Nov 13 2019
Equals c*(2*c+1), where c = A153415 (follows from the identity Sum_{n=-oo..oo} 1/L(n^2) = (Sum_{n=-oo..oo} 1/L(2*n))^2, see Bruckman, 1982). - Amiram Eldar, Jan 27 2022
EXAMPLE
1.207291996985747074417204...
MATHEMATICA
f[n_] := f[n] = RealDigits[ Sum[ 1/LucasL[k]^2, {k, 1, n}], 10, 100] // First; f[n=100]; While[f[n] != f[n-100], n = n+100]; f[n] (* Jean-François Alcover, Feb 13 2013 *)
CROSSREFS
Cf. A000032, A001254 (squares of Lucas numbers).
Sequence in context: A300704 A300702 A353273 * A196767 A307216 A011343
KEYWORD
cons,easy,nonn
AUTHOR
Jonathan Vos Post, Apr 04 2005
STATUS
approved