A100063 A Chebyshev transform of Jacobsthal numbers.

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1

Offset

0,4

Comments

A Chebyshev transform of A001045(n+1): if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))*A(x/(1+x^2)).

Also decimal expansion of 1111/9990. - Elmo R. Oliveira, Feb 18 2024

Links

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

Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Formula

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

a(n) = n*Sum_{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*A001045(n-2k+1)/(n-k).

Multiplicative with a(3^e) = 2, a(p^e) = 1 otherwise. - David W. Wilson, Jun 11 2005

Dirichlet g.f.: zeta(s)*(1+1/3^s). Dirichlet convolution of A154272 and A000012. - R. J. Mathar, Feb 07 2011

a(n) = 2 if n == 0 (mod 3) and n > 0, and a(n) = 1 otherwise. - Amiram Eldar, Nov 01 2022

a(n) = gcd(Fibonacci(n), Lucas(n)) = gcd(A000045(n), A000032(n)), for n >= 1. - Amiram Eldar, Jul 10 2023

Example

G.f. = 1 + x + x^2 + 2*x^3 + x^4 + x^5 + 2*x^6 + x^7 + x^8 + 2*x^9 + ... - Michael Somos, Feb 20 2024

Mathematica

PadRight[{1}, 120, {2, 1, 1}] (* or *) LinearRecurrence[{0, 0, 1}, {1, 1, 1, 2}, 120] (* Harvey P. Dale, Jul 08 2015 *)
a[ n_] := If[n<1, Boole[n==0], {2, 1, 1}[[1+Mod[n, 3]]]]; (* Michael Somos, Feb 20 2024 *)

Prog

(PARI) my(x='x+O('x^50)); Vec((1+x)(1+x^2)/(1-x^3)) \\ G. C. Greubel, May 03 2017
(PARI) {a(n) = if(n<1, n==0, [2, 1, 1][n%3+1])}; /* Michael Somos, Feb 20 2024 */

Crossrefs

Cf. A000012, A001045, A100051, A061347, A057079, A154272.

Cf. A000032, A000045.

Sequence in context: A281727 A122876 A131713 * A057079 A132419 A131556

Adjacent sequences: A100060 A100061 A100062 * A100064 A100065 A100066

Keyword

easy,nonn,mult

Author

Paul Barry, Nov 02 2004

Status

approved