A083216 Fibonacci-like sequence of composite numbers with a(0) = 20615674205555510, a(1) = 3794765361567513.

20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536, 52615644495813559, 80820849424504095, 133436493920317654, 214257343344821749, 347693837265139403, 561951180609961152, 909645017875100555, 1471596198485061707, 2381241216360162262

Offset: 0

Author

Harry J. Smith, Apr 23 2003

Keywords

• nonn
• easy

Comments

a(0) = 20615674205555510, a(1) = 3794765361567513. 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 Herbert S. Wilf in 1990.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..4709 (terms 0..1000 from Alois P. Heinz)
• 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, A000045, A083103, A083104, A083105, A082411, A221286.

Programs

Maple

a:= n-> (<<0|1>, <1|1>>^n. <<20615674205555510, 3794765361567513>>)[1, 1]:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 04 2013

Mathematica

LinearRecurrence[{1,1},{20615674205555510,3794765361567513},25] (* Paolo Xausa, Nov 07 2023 *)

PARI

Vec((20615674205555510-16820908843987997*x)/(1-x-x^2)+O(x^9)) \\ Charles R Greathouse IV, Sep 23 2012

Formula

a(n) = a(n-1) + a(n-2) for n>1.

G.f.: (20615674205555510-16820908843987997*x)/(1-x-x^2).