A105394 Decimal expansion of sum of reciprocals of squares of Lucas numbers.
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
Examples
1.207291996985747074417204...
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.
- Index entries for transcendental numbers
Programs
-
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 *)
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
Comments