A014985 a(n) = (1 - (-4)^n)/5.

1, -3, 13, -51, 205, -819, 3277, -13107, 52429, -209715, 838861, -3355443, 13421773, -53687091, 214748365, -858993459, 3435973837, -13743895347, 54975581389, -219902325555, 879609302221, -3518437208883, 14073748835533

Offset: 1

Author

Olivier Gérard

Keywords

• sign
• easy

Comments

q-integers for q=-4.

In Penrose's book, presented as partial sums of the series for 1/(1-x^2) evaluated at x=2. - Olivier Gérard, May 22 2009

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=2, a(n-1)=(-1)^n*charpoly(A,1). - Milan Janjic, Jan 27 2010

References

• Roger Penrose, "The Road to Reality, A complete guide to the Laws of the Universe", Jonathan Cape, London, 2004, pages 79-80. - Olivier Gérard, May 22 2009

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (-3,4).

Crossrefs

Cf. A077925, A014983, A014986, A014987, A014989-A014994.

Programs

Magma

I:=[1, -3]; [n le 2 select I[n] else -3*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 21 2012

Maple

a:=n->sum ((-4)^j, j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Dec 16 2008

Mathematica

LinearRecurrence[{-3, 4}, {1, -3}, 50] (* or *) CoefficientList[ Series[ 1/((1-x)*(1+4*x)), {x,0,30}], x] (* Vincenzo Librandi, Oct 21 2012 *)
(1-(-4)^Range[30])/5 (* Harvey P. Dale, Oct 06 2024 *)

PARI

a(n)=(1-(-4)^n)/5 \\ Charles R Greathouse IV, Sep 24 2015

SageMath

[gaussian_binomial(n,1,-4) for n in range(1,24)] # Zerinvary Lajos, May 28 2009

Formula

a(n) = a(n-1) + q^{(n-1)} = {(q^n - 1) / (q - 1)}, with q=-4.

From Paul Barry, Jan 12 2007: (Start)

G.f.: x/(1+3*x-4*x^2).

a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*4^k*(-3)^(n-2k). (End)

a(n) = -3*a(n-1) +4*a(n-2). - Vincenzo Librandi, Oct 21 2012

Extensions

G.f. adapted to the offset by Vincenzo Librandi, Oct 21 2012

Better name from Ralf Stephan, Jul 14 2013