A307991 Fibonacci numbers of the form k^2 - k - 1 with integer k.

1, 5, 55, 89

Offset: 1

Author

Amiram Eldar, May 09 2019

Keywords

• nonn
• bref
• fini
• full

Comments

The corresponding values of k are 2, 3, 8, 10.

Intersection of A000045 and A028387.

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.

Examples

			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.
		

References

• Fenton Stancliff, A curious property of a_11, Scripta Math., Vol. 19 (1953), p. 126.

Links

• B. M. M. de Weger, A Curious Property of the Eleventh Fibonacci Number, The Rocky Mountain Journal of Mathematics, Vol. 25, No. 3 (1995), pp. 977-994.

Crossrefs

Cf. A000045, A021093, A028387, A039595, A165900, A227875.

Programs

Mathematica

Select[Fibonacci[Range[2, 20]], IntegerQ[Sqrt[4# + 5]] &]