This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A307991 #6 May 10 2019 22:42:53
%S A307991 1,5,55,89
%N A307991 Fibonacci numbers of the form k^2 - k - 1 with integer k.
%C A307991 The corresponding values of k are 2, 3, 8, 10.
%C A307991 Intersection of A000045 and A028387.
%C A307991 Also Fibonacci numbers whose reciprocals equal to Sum_{i>=1} F(i)/k^(i+1), where F(i) is the i-th Fibonacci number.
%C A307991 de Weger proved that there are no other terms.
%D A307991 Fenton Stancliff, A curious property of a_11, Scripta Math., Vol. 19 (1953), p. 126.
%H A307991 B. M. M. de Weger, <a href="https://doi.org/10.1216/rmjm/1181072199">A Curious Property of the Eleventh Fibonacci Number</a>, The Rocky Mountain Journal of Mathematics, Vol. 25, No. 3 (1995), pp. 977-994.
%e A307991 89 is in the sequence since 89 = 10^2 - 10 - 1 or equivalently 1/89 = 1/10^2 + 1/10^3 + 2/10^4 + 3/10^5 + 5/10^6 + ... This is why the first digits of the decimal expansion of 1/89 = 0.011235... are the first terms of the Fibonacci sequence.
%t A307991 Select[Fibonacci[Range[2, 20]], IntegerQ[Sqrt[4# + 5]] &]
%Y A307991 Cf. A000045, A021093, A028387, A039595, A165900, A227875.
%K A307991 nonn,bref,fini,full
%O A307991 1,2
%A A307991 _Amiram Eldar_, May 09 2019