OFFSET
1,2
COMMENTS
Second diagonal of array A143979, which counts certain unit squares in a lattice. First diagonal: A030511.
Convolution of A042965 with A000012, convolution of A131534 with A000027, and convolution of A106510 with A000217. - L. Edson Jeffery, Jan 24 2015
From Miquel A. Fiol, Aug 31 2024: (Start)
a(n+1) is the maximum number N of vertices of a circulant digraph with steps +-s1, s2, and diameter n.
Depending on the value of n, the following table shows the values of N, s1, and s2:
n | 3*r | 3*r-1 | 3*r-2 |
N | 6*r^2+6*r+1 | 6*r^2+2*r | 6*r^2-2*r |
s1 | 1 | r | r |
s2 | 6*r+3 | 3*r+1 | 3*r-1 |
(End)
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Bruno Berselli, Illustration of the initial terms.
P. Morillo and Miquel A. Fiol, El diámetro de ciertos digrafos circulantes de triple paso, Stochastica X (1986), no. 3, 233-249.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).
FORMULA
From R. J. Mathar, Oct 05 2009: (Start)
G.f.: x*(1 + x)^2/((1 + x + x^2)*(1-x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5). (End)
a(n) = Sum_{k=1..(n+1)} A042965(k). - Klaus Purath, May 23 2020
From G. C. Greubel, May 27 2020: (Start)
a(n) = (ChebyshevU(n, -1/2) - ChebyshevU(n-1, -1/2) + (6*n^2 + 6*n -1))/9.
a(n) = (JacobiSymbol(n+1, 3) - JacobiSymbol(n, 3) + (6*n^2 + 6*n -1))/9.
Sum_{n>=1} 1/a(n) = 3/2 + (tan(Pi/(2*sqrt(3)))-1)*Pi/(2*sqrt(3)). - Amiram Eldar, Sep 27 2022
E.g.f.: exp(-x/2)*(exp(3*x/2)*(6*x^2 + 12*x - 1) + cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Apr 05 2023
MAPLE
A143978:= n-> (6*n*(n+1) -1 + `mod`(n+2, 3) - `mod`(n+1, 3))/9;
seq(A143978(n), n=1..60); # G. C. Greubel, May 27 2020
MATHEMATICA
Table[(6*n^2 +6*n -1 + Mod[n+2, 3] - Mod[n+1, 3])/9, {n, 60}] (* G. C. Greubel, May 27 2020 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 06 2008
STATUS
approved