A111734 Expansion of (1-x)*(2*x^2+2*x+1) / ((x^2-x-1)*(x^2+x+1)).

-1, 1, -1, 3, -6, 10, -15, 23, -37, 61, -100, 162, -261, 421, -681, 1103, -1786, 2890, -4675, 7563, -12237, 19801, -32040, 51842, -83881, 135721, -219601, 355323, -574926, 930250, -1505175, 2435423, -3940597, 6376021, -10316620, 16692642, -27009261, 43701901, -70711161, 114413063

Offset: 1

Author

Creighton Dement, Nov 18 2005

Keywords

• easy
• sign

Comments

Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[(.5'j + .5'k + .5j' + .5k' + .5'ij' - .5'ik' + .5'ji' + .5'ki')*(.5'i + .5'j + .5'k + .5e)]

References

• Creighton Dement, Floretion Integer Sequences (work in progress).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,-1,0,1).

Crossrefs

Cf. A000204, A007039.

Programs

Mathematica

CoefficientList[Series[(1-x)(2x^2+2x+1)/((x^2-x-1)(x^2+x+1)),{x,0,60}],x] (* or *) LinearRecurrence[{-2,-1,0,1},{-1,1,-1,3},60] (* Harvey P. Dale, Sep 01 2021 *)

PARI

Vec(-x*(1 - x)*(1 + 2*x + 2*x^2) / ((1 + x - x^2)*(1 + x + x^2)) + O(x^40)) \\ Colin Barker, May 18 2019

Formula

a(n) = ((-1)^n/2)*A007039(n); a(n) + a(n+1) + a(n+2) = ((-1)^n)*A000204(n).

a(n) = (-1)^n*(n*sum((binomial(n-2*m,2*m))/(n-2*m),m,0,floor((n-1)/2))). - Vladimir Kruchinin, Mar 10 2013

a(n) = -2*a(n-1) - a(n-2) + a(n-4) for n>4. - Colin Barker, May 18 2019

E.g.f.: exp(-x/2)*(cos(sqrt(3)*x/2) + cosh(sqrt(5)*x/2) + sqrt(3)*sin(sqrt(3)*x/2) - sqrt(5)*sinh(sqrt(5)*x/2))/2. - Stefano Spezia, Aug 03 2022