A167373 Expansion of (1+x)*(3*x+1)/(1+x+x^2).

1, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1

Offset: 0

Author

Jamel Ghanouchi, Nov 02 2009

Keywords

• sign
• easy

Comments

Bisection of A138034.

Also row 2n of A137276 or A135929.

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 22.

Links

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

Crossrefs

Cf. A135929, A138034, A137276.

Programs

Maple

A167373 := proc(n)
    option remember;
    if n < 4 then
        op(n+1,[1,3,-1,-2]) ;
    else
        procname(n-3) ;
    end if;
end proc:
seq(A167373(n),n=0..20) ; # R. J. Mathar, Feb 06 2020

Mathematica

CoefficientList[Series[(1 + x)*(3*x + 1)/(1 + x + x^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 12 2016 *)
LinearRecurrence[{-1,-1},{1,3,-1},120] (* Harvey P. Dale, Apr 05 2023 *)

Formula

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

a(n) = a(n-3), n>4.

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

a(n) = 4*sin(2*n*Pi/3)/sqrt(3)-2*cos(2*n*Pi/3) for n>0 with a(0)=1. - Wesley Ivan Hurt, Jun 12 2016

Extensions

Edited by R. J. Mathar, Nov 03 2009

Further edited and extended by Simon Plouffe, Nov 23 2009

Recomputed by N. J. A. Sloane, Dec 20 2009