A084086 a(n) = Fibonacci(2*n+1) + 2*Fibonacci(2*n-1) - 2^n - [n = 0], where [b] is the Iverson bracket of b.

1, 2, 5, 15, 44, 125, 347, 948, 2561, 6863, 18284, 48501, 128243, 338276, 890681, 2341959, 6151580, 16145549, 42350603, 111037332, 291023537, 762557567, 1997697740, 5232632805, 13704394979, 35888940740, 93979204457, 246082227063, 644334585596, 1687055747453

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-7,2).

Formula

Length of lists created by n substitutions, in Mathematica syntax:

k -> Range[ -1-Abs[k] + MoebiusMu[Abs[k]], k + 1 + MoebiusMu[Abs[k]], 2], starting with {-1}.

Equivalent to -10 -> {-10,-8}; -9 -> {-10,-8}; -8 -> {-9,-7}; -7 -> {-9,-7}; -5 -> {-7,-5}; -3 -> {-5,-3}; -1 -> {-1,1}; 1 -> {-1,1,3}; 3 -> {-5,-3,-1,1,3}.

G.f.: (1 - 3*x + 2*x^2 + 2*x^3)/(1 - 5*x + 7*x^2 - 2*x^3).

From G. C. Greubel, Oct 15 2022: (Start)

a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3), for n >= 4.

a(n) = 3*ChebyshevU(n, 3/2) - 5*ChebyshevU(n-1, 3/2) - 2^n - [n=0].

a(n) = Fibonacci(2*n+1) + 2*Fibonacci(2*n-1) - 2^n - [n=0]. (End)

Example

Length of {-1}, {-1,1}, {-1,1,-1,1,3}, {-1,1,-1,1,3,-1,1,-1,1,3,-5,-3,-1,1,3} is 1, 2, 5, 15, respectively.

Maple

F := n -> combinat:-fibonacci(n):
a := n -> F(2*n+1) + 2*F(2*n-1) - 2^n - ifelse(n=0, 1, 0):
seq(a(n), n = 0..29); # Peter Luschny, Oct 16 2022

Mathematica

Length/@Flatten/@NestList[ # /. k_Integer:>Range[ -1-Abs[k]+MoebiusMu[Abs[k]], k+1+MoebiusMu[Abs[k]], 2]&, {-1}, 8]
(* Second program *)
LinearRecurrence[{5, -7, 2}, {1, 2, 5, 15}, 40] (* G. C. Greubel, Oct 15 2022 *)

Prog

(Magma) [1] cat [Fibonacci(2*n+1) +2*Fibonacci(2*n-1) -2^n: n in [1..40]]; // G. C. Greubel, Oct 15 2022
(SageMath) [fibonacci(2*n+1) +2*fibonacci(2*n-1) -2^n -int(n==0) for n in range(41)] # G. C. Greubel, Oct 15 2022

Crossrefs

Cf. A001519, A001906.

Sequence in context: A304201 A215448 A094176 * A307259 A292524 A071742

Adjacent sequences: A084083 A084084 A084085 * A084087 A084088 A084089

Keyword

nonn,easy

Author

Wouter Meeussen, May 11 2003

Extensions

New name using a formula of G. C. Greubel by Peter Luschny, Oct 16 2022

Status

approved