OFFSET
1,2
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).
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
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 *)
PROG
(Sage) [gaussian_binomial(n, 1, -4) for n in range(1, 24)] # - Zerinvary Lajos, May 28 2009
(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
(PARI) a(n)=(1-(-4)^n)/5 \\ Charles R Greathouse IV, Sep 24 2015
CROSSREFS
KEYWORD
sign,easy
AUTHOR
EXTENSIONS
G.f. adapted to the offset by Vincenzo Librandi, Oct 21 2012
Better name from Ralf Stephan, Jul 14 2013
STATUS
approved