A082411 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2), with initial terms 407389224418, 76343678551.

407389224418, 76343678551, 483732902969, 560076581520, 1043809484489, 1603886066009, 2647695550498, 4251581616507, 6899277167005, 11150858783512, 18050135950517, 29200994734029, 47251130684546, 76452125418575, 123703256103121, 200155381521696, 323858637624817

Offset: 0

Author

Harry J. Smith, Apr 23 2003

Keywords

• nonn
• easy

Comments

a(0) = 407389224418, a(1) = 76343678551. This is a second-order linear recurrence sequence with a(0) and a(1) coprime that does not contain any primes. It was found by John Nicol in 1999.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Arturas Dubickas, Aivaras Novikas and Jonas Šiurys, A binary linear recurrence sequence of composite numbers, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1737-1749.
• R. L. Graham, A Fibonacci-Like sequence of composite numbers, Math. Mag. 37 (1964) 322-324.
• D. Ismailescu and 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
• J. W. Nicol, A Fibonacci-like sequence of composite numbers, The Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R44.
• Herbert S. Wilf, Letters to the Editor Math. Mag. 63, 284, 1990.
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Cf. A000032 (Lucas numbers), A000045 (Fibonacci numbers), A083103, A083104, A083105, A083216, A221286.

Programs

Maple

a:= n-> (<<0|1>, <1|1>>^n. <<407389224418, 76343678551>>)[1, 1]:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 04 2013

Mathematica

LinearRecurrence[{1,1},{407389224418,76343678551},25] (* Paolo Xausa, Nov 07 2023 *)

Formula

G.f.: (407389224418-331045545867*x)/(1-x-x^2). [Colin Barker, Jun 19 2012]

Extensions

Name clarified by Robert C. Lyons, Feb 07 2025