cp's OEIS Frontend

A105394 Decimal expansion of sum of reciprocals of squares of Lucas numbers.

%I A105394 #43 Feb 16 2025 08:32:57
%S A105394 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,
%T A105394 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,
%U A105394 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
%N A105394 Decimal expansion of sum of reciprocals of squares of Lucas numbers.
%C A105394 This constant is transcendental (Duverney et al., 1997). - _Amiram Eldar_, Oct 30 2020
%D A105394 Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Wiley, 1987, p. 97.
%H A105394 Richard André-Jeannin, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5686125p/f9.image">Irrationalité de la somme des inverses de certaines suites récurrentes</a>, C. R. Acad. Sci. Paris Ser. I Math., Vol. 308, No. 19 (1989), pp. 539-541.
%H A105394 Paul S. Bruckman,, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/20-4/advanced20-4.pdf">Problem H-347</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 20, No. (1982), p. 372; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/22-1/advanced22-1.pdf">It All Adds Up</a>, Solution to Problem H-347 by the proposer, ibid., Vol. 22, No. 1 (1984), pp. 94-96.
%H A105394 Daniel Duverney, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa, <a href="http://doi.org/10.3792/pjaa.73.140">Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers</a>, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 7 (1997), pp. 140-142.
%H A105394 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>.
%H A105394 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>.
%H A105394 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ReciprocalFibonacciConstant.html">Reciprocal Fibonacci Constant</a>.
%H A105394 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A105394 Equals Sum_{n >= 1} 1/L(n)^2.
%F A105394 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
%F A105394 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
%e A105394 1.207291996985747074417204...
%t A105394 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 *)
%Y A105394 Cf. A000032, A001254 (squares of Lucas numbers).
%Y A105394 Cf. A079586, A093540, A105393, A153415.
%K A105394 cons,easy,nonn
%O A105394 1,2
%A A105394 _Jonathan Vos Post_, Apr 04 2005