A112455 a(n) = -a(n-2) - a(n-3).

-3, 0, 2, 3, -2, -5, -1, 7, 6, -6, -13, 0, 19, 13, -19, -32, 6, 51, 26, -57, -77, 31, 134, 46, -165, -180, 119, 345, 61, -464, -406, 403, 870, 3, -1273, -873, 1270, 2146, -397, -3416, -1749, 3813, 5165, -2064, -8978, -3101, 11042, 12079, -7941

Offset: 0

Author

Anthony C Robin, Dec 13 2005

Keywords

• sign
• easy

Comments

This sequence resembles the Perrin sequence, A001608. Like many such sequences with a(1)=0, any prime p divides a(p). The first pseudoprime (composite n divides a(n)) is 121.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (0,-1,-1).

Crossrefs

Cf. A001608, A112458.

Programs

GAP

a:=[-3,0,2];; for n in [4..60] do a[n]:=-a[n-2]-a[n-3]; od; a; # G. C. Greubel, May 19 2019

Magma

I:=[-3,0,2]; [n le 3 select I[n] else -Self(n-2) -Self(n-3): n in [1..60]]; // G. C. Greubel, May 19 2019

Maple

A112455 := proc(n)
    option remember ;
    if n <= 2 then
        op(n+1,[-3,0,2]) ;
    else
        -procname(n-2)-procname(n-3) ;
    end if;
end proc: # R. J. Mathar, Feb 18 2024

Mathematica

Table[ -Tr[MatrixPower[{{0, 0, -1}, {1, 0, -1}, {0, 1, 0}}, n]], {n, 1, 60}] (* Artur Jasinski, Jan 10 2007 *)
LinearRecurrence[{0,-1,-1}, {-3,0,2}, 60] (* G. C. Greubel, May 19 2019 *)

PARI

Vec(-(3+x^2)/(1+x^2+x^3)+O(x^60)) \\ Charles R Greathouse IV, May 15 2013

Sage

(-(3+x^2)/(1+x^2+x^3)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, May 19 2019

Formula

a(n) = - trace({{0, 0, -1}, {1, 0, -1}, {0, 1, 0}})^n. - Artur Jasinski, Jan 10 2007

From R. J. Mathar, Oct 24 2009: (Start)

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

a(n) = -3*A077962(n) - A077962(n-2). (End)

a(n) = (-1)^(n+1)*(A001609(n)^2 - A001609(2*n))/2. - Greg Dresden, Apr 14 2023

Extensions

Edited by Don Reble, Jan 25 2006