

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
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OFFSET

1,2


COMMENTS

qintegers for q=4.
In Penrose's book, presented as partial sums of the series for 1/(1x^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,i1]=1, and A[i,j]=0 otherwise. Then, for n>=2, a(n1)=(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 7980.  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(n1) + q^{(n1)} = {(q^n  1) / (q  1)}, with q=4.
From Paul Barry, Jan 12 2007: (Start)
G.f.: x/(1+3*x4*x^2).
a(n) = Sum_{k=0..floor(n/2)} C(nk,k)*4^k*(3)^(n2k). (End)
a(n) = 3*a(n1) +4*a(n2).  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/((1x)*(1+4*x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 21 2012 *)


PROG

(Sage) [gaussian_binomial(n, 1, 4) for n in xrange(1, 24)] #  Zerinvary Lajos, May 28 2009
(MAGMA) I:=[1, 3]; [n le 2 select I[n] else 3*Self(n1)+4*Self(n2): 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, A014989A014994.
Sequence in context: A301458 A244784 A197074 * A015521 A270913 A323266
Adjacent sequences: A014982 A014983 A014984 * A014986 A014987 A014988


KEYWORD

sign,easy


AUTHOR

Olivier Gérard


EXTENSIONS

G.f. adapted to the offset by Vincenzo Librandi, Oct 21 2012
Better name from Ralf Stephan, Jul 14 2013


STATUS

approved



