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 A333208 a(n) is the numerator of Sum_{i >= 0} 1/(Lucas(i)*Lucas(i+2n)), with Lucas(i) as defined in A000032. 1
 1, 7, 551, 48091, 2148268, 87644575267, 161577754532123, 36595152483523582367, 16965509829762630129638831, 173107561150078104051618631740949, 1108595900580419409151086339986148307, 2608169750203411467722731179728125652086612772 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The denominators are given in A333209. See A333088 and A333089 for similar fractions for infinite sums of Fibonacci numbers. Sum_{i >= 0} 1/(Lucas(i)*Lucas(i+2n)) is a fraction for n > 0. Sum_{i >= 0} 1/Lucas(i)^2 = 1/4 + A105394, i.e., the n = 0 case, is believed to be transcendental. LINKS FORMULA a(n) = numerator of (1/Fibonacci(2n)) * Sum_{i=1..n} 1/(Lucas(2i-2)*Lucas(2i-1)). Lim_{n -> inf} (a(n)/A333209(n)) / (a(n-1)/A333209(n-1)) = 1 - 1/phi = 1/phi^2 = A132318. The following generalization holds: (Start) Let H_(a,b) (n) be defined by H_(a,b) (0) = a, H_(a,b) (1) = b and H_(a,b) (n) = H_(a,b) (n-1) + H_(a,b) (n-2) for n > 1, then Sum_{i >= 0} 1/(H_(a,b) (i)*H_(a,b) (i+2n)) = (1/Fibonacci(2n)) * Sum_{i=1..n} 1/(H_(a,b) (2i-2)*H_(a,b) (2i-1)) for n > 0, and are thus all fractions. Specially, H_(0,1) are the Fibonacci numbers A000045, H_(2,1) as here, are the Lucas numbers A000032, and H_(3,1) are the Pibonacci numbers A104449. (End) EXAMPLE These infinite sums begin: 1/2, 7/36, 551/7392, ... MATHEMATICA a[n_] := Numerator[Sum[1/(LucasL[2 i - 2]*LucasL[2 i - 1]), {i, 1, n}]/Fibonacci[2 n]]; Array[a, 12] (* Amiram Eldar, Mar 11 2020 *) CROSSREFS Cf. A000032, A105394, A132318, A333088, A333089, A333209. Sequence in context: A126196 A182433 A329368 * A093169 A159029 A308582 Adjacent sequences:  A333205 A333206 A333207 * A333209 A333210 A333211 KEYWORD nonn AUTHOR A.H.M. Smeets, Mar 11 2020 STATUS approved

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Last modified August 11 04:00 EDT 2020. Contains 336421 sequences. (Running on oeis4.)