OFFSET
0,3
COMMENTS
Binomial transform of A099462.
From Paul Barry, Feb 07 2006: (Start)
Partial sums are A113300. (End)
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Meng-Han Wu, Henryk A. Witek, Rafał Podeszwa, Clar Covers and Zhang-Zhang Polynomials of Zigzag and Armchair Carbon Nanotubes, MATCH Commun. Math. Comput. Chem. (2025) Vol. 93, 415-462. See p. 437.
Index entries for linear recurrences with constant coefficients, signature (3,1,1).
FORMULA
G.f.: x*(1-x)/(1-3*x-x^2-x^3).
a(n) = Sum_{k=0..n} binomial(n, k)*Sum_{j=0..floor((k-1)/2)} binomial(j, k-2*j-1)*4^j.
From Paul Barry, Feb 07 2006: (Start)
a(n) = 3*a(n-1) + a(n-2) + a(n-3).
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(2*k, n-k-j)*C(n-k, j)*2^(n-k-j). (End)
a(n)/a(n-1) tends to 3.38297576..., the square of the tribonacci constant A058265. - Gary W. Adamson, Feb 28 2006
If p[1]=2, p[2]=3, p[i]=4, (i>2), and if A is Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n+1) = det A. - Milan Janjic, May 02 2010
MATHEMATICA
LinearRecurrence[{3, 1, 1}, {0, 1, 2}, 30] (* or *) Join[{0}, Mean/@ Partition[ LinearRecurrence[ {1, 1, 1}, {1, 1, 1}, 60], 2]] (* Harvey P. Dale, Apr 02 2012 *)
PROG
(Magma) [n le 3 select (n-1) else 3*Self(n-1) +Self(n-2) +Self(n-3): n in [1..31]]; // G. C. Greubel, Nov 20 2021
(Sage)
def A184883(n, k): return simplify( hypergeometric([-k, 2*(k-n)], [1], 2) )
[0]+[A099463(n-1) for n in (1..40)] # G. C. Greubel, Nov 20 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 16 2004
STATUS
approved