A083103 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2), with initial terms 1786772701928802632268715130455793, 1059683225053915111058165141686995.

1786772701928802632268715130455793, 1059683225053915111058165141686995, 2846455926982717743326880272142788, 3906139152036632854385045413829783, 6752595079019350597711925685972571, 10658734231055983452096971099802354

Offset: 0

Author

Harry J. Smith, Apr 22 2003

Keywords

• nonn
• easy

Comments

a(0) = 1786772701928802632268715130455793, a(1) = 1059683225053915111058165141686995. This is the second-order linear recurrence sequence with a(0) and a(1) coprime that R. L. Graham in 1964 stated did not contain any primes. It has not been verified. Graham made a mistake in the calculation that was corrected by D. E. Knuth in 1990.

References

• P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 178.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..4617
• R. L. Graham, A Fibonacci-Like sequence of composite numbers, Math. Mag. 37 (1964) 322-324
• D. Ismailescu, J. Son, A New Kind of Fibonacci-Like Sequence of Composite Numbers, J. Int. Seq. 17 (2014) # 14.8.2.
• Tanya Khovanova, Recursive Sequences
• D. E. Knuth, A Fibonacci-Like sequence of composite numbers, Math. Mag. 63 (1) (1990) 21-25
• Carlos Rivera, Problem 31. Fibonacci- all composites sequence, The Prime Puzzles and Problems Connection.
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Cf. A000032 (Lucas numbers), A000045 (Fibonacci numbers), A083104, A083105.

Programs

Mathematica

LinearRecurrence[{1,1},{1786772701928802632268715130455793, 1059683225053915111058165141686995},70] (* Harvey P. Dale, Oct 17 2011 *)

Formula

G.f.: (1786772701928802632268715130455793-727089476874887521210549988768798*x)/(1-x-x^2). [Colin Barker, Jun 19 2012]

Extensions

Name clarified by Robert C. Lyons, Feb 07 2025