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 A014985 a(n) = (1 - (-4)^n)/5. 20
 1, -3, 13, -51, 205, -819, 3277, -13107, 52429, -209715, 838861, -3355443, 13421773, -53687091, 214748365, -858993459, 3435973837, -13743895347, 54975581389, -219902325555, 879609302221, -3518437208883, 14073748835533 (list; graph; refs; listen; history; text; internal format)
 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 *) 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 Cf. A077925, A014983, A014986, A014987, A014989-A014994. Sequence in context: A301458 A244784 A197074 * A015521 A270913 A323266 Adjacent sequences:  A014982 A014983 A014984 * A014986 A014987 A014988 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

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Last modified January 24 07:45 EST 2021. Contains 340398 sequences. (Running on oeis4.)