A083105 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2), with initial terms 62638280004239857, 49463435743205655.

62638280004239857, 49463435743205655, 112101715747445512, 161565151490651167, 273666867238096679, 435232018728747846, 708898885966844525, 1144130904695592371, 1853029790662436896, 2997160695358029267, 4850190486020466163, 7847351181378495430

Offset: 0

Author

Harry J. Smith, Apr 23 2003

Keywords

• nonn
• easy

Comments

a(0) = 62638280004239857, a(1) = 49463435743205655. 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 D. E. Knuth in 1990.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..4705
• 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.
• 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), A083103, A083104, A083216, A082411, A221286.

Programs

Maple

a:= n-> (<<0|1>, <1|1>>^n. <<62638280004239857, 49463435743205655>>)[1, 1]:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 20 2021

Mathematica

LinearRecurrence[{1,1},{62638280004239857,49463435743205655},20] (* Paolo Xausa, Nov 07 2023 *)

Formula

G.f.: (62638280004239857-13174844261034202*x)/(1-x-x^2). [Colin Barker, Jun 19 2012]

Extensions

Name clarified by Robert C. Lyons, Feb 07 2025