A099842 Expansion of (1-x)/(1 + 6*x - 3*x^2).

1, -7, 45, -291, 1881, -12159, 78597, -508059, 3284145, -21229047, 137226717, -887047443, 5733964809, -37064931183, 239591481525, -1548743682699, 10011236540769, -64713650292711, 418315611378573, -2704034619149571, 17479154549033145, -112987031151647583

Offset: 0

Author

Paul Barry, Oct 27 2004

Keywords

• easy
• sign

Comments

A transformation of x/(1-2*x-2*x^2).

The g.f. is the transform of the g.f. of A002605 under the mapping G(x) -> (-1/(1+x))*G((x-1)/(x+1)). In general this mapping transforms x/(1-k*x-k*x^2) into (1-x)/(1+2*(k+1)*x-(2*k-1)*x^2).

For n >= 1, |a(n)| equals the numbers of words of length n-1 on alphabet {0,1,...,6} containing no subwords 00, 11, 22, 33. - Milan Janjic, Jan 31 2015

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-6,3).

Crossrefs

Cf. A002605, A090018.

Programs

Magma

[n le 2 select (-7)^(n-1) else -6*Self(n-1) +3*Self(n-2): n in [1..31]]; // G. C. Greubel, Oct 10 2022

Mathematica

LinearRecurrence[{-6,3}, {1,-7}, 31] (* G. C. Greubel, Oct 10 2022 *)
CoefficientList[Series[(1-x)/(1+6x-3x^2),{x,0,40}],x] (* Harvey P. Dale, Aug 31 2024 *)

SageMath

A099842 = BinaryRecurrenceSequence(-6,3,1,-7)
[A099842(n) for n in range(31)] # G. C. Greubel, Oct 10 2022

Formula

G.f.: (1-x)/(1+6*x-3*x^2).

a(n) = (1/2 - sqrt(3)/3)*(-3 + 2*sqrt(3))^n + (1/2 + sqrt(3)/3)*(-3 - 2*sqrt(3))^n.

a(n) = (-1)^n*Sum_{k=0..n} binomial(n, k)(-1)^(n-k)*A002605(2k+2)/2.

a(n) = (-1)^n*(A090018(n) + A090018(n-1)). - R. J. Mathar, Apr 07 2022