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A307991
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Fibonacci numbers of the form k^2 - k - 1 with integer k.
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0
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OFFSET
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1,2
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COMMENTS
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The corresponding values of k are 2, 3, 8, 10.
Also Fibonacci numbers whose reciprocals equal to Sum_{i>=1} F(i)/k^(i+1), where F(i) is the i-th Fibonacci number.
de Weger proved that there are no other terms.
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REFERENCES
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Fenton Stancliff, A curious property of a_11, Scripta Math., Vol. 19 (1953), p. 126.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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Select[Fibonacci[Range[2, 20]], IntegerQ[Sqrt[4# + 5]] &]
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CROSSREFS
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KEYWORD
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nonn,bref,fini,full
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AUTHOR
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STATUS
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approved
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