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A143978 a(n) = floor(2*n*(n+1)/3).

%I A143978 #56 Aug 12 2025 11:53:01
%S A143978 1,4,8,13,20,28,37,48,60,73,88,104,121,140,160,181,204,228,253,280,
%T A143978 308,337,368,400,433,468,504,541,580,620,661,704,748,793,840,888,937,
%U A143978 988,1040,1093,1148,1204,1261,1320,1380,1441,1504,1568,1633,1700,1768,1837
%N A143978 a(n) = floor(2*n*(n+1)/3).
%C A143978 Second diagonal of array A143979, which counts certain unit squares in a lattice. First diagonal: A030511.
%C A143978 Convolution of A042965 with A000012, convolution of A131534 with A000027, and convolution of A106510 with A000217. - _L. Edson Jeffery_, Jan 24 2015
%C A143978 From _Miquel A. Fiol_, Aug 31 2024: (Start)
%C A143978 a(n+1) is the maximum number N of vertices of a circulant digraph with steps +-s1, s2, and diameter n.
%C A143978 Depending on the value of n, the following table shows the values of N, s1, and s2:
%C A143978  n |     3*r     |   3*r-1   |   3*r-2   |
%C A143978  N | 6*r^2+6*r+1 | 6*r^2+2*r | 6*r^2-2*r |
%C A143978 s1 |      1      |     r     |     r     |
%C A143978 s2 |    6*r+3    |   3*r+1   |   3*r-1   |
%C A143978 (End)
%H A143978 G. C. Greubel, <a href="/A143978/b143978.txt">Table of n, a(n) for n = 1..1000</a>
%H A143978 Bruno Berselli, <a href="/A143978/a143978.jpg">Illustration of the initial terms</a>.
%H A143978 P. Morillo and Miquel A. Fiol, <a href="http://hdl.handle.net/2117/2109">El diámetro de ciertos digrafos circulantes de triple paso</a>, Stochastica X (1986), no. 3, 233-249.
%H A143978 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F A143978 From _R. J. Mathar_, Oct 05 2009: (Start)
%F A143978 G.f.: x*(1 + x)^2/((1 + x + x^2)*(1-x)^3).
%F A143978 a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5). (End)
%F A143978 a(n) = Sum_{k=1..(n+1)} A042965(k). - _Klaus Purath_, May 23 2020
%F A143978 From _G. C. Greubel_, May 27 2020: (Start)
%F A143978 a(n) = (ChebyshevU(n, -1/2) - ChebyshevU(n-1, -1/2) + (6*n^2 + 6*n -1))/9.
%F A143978 a(n) = (JacobiSymbol(n+1, 3) - JacobiSymbol(n, 3) + (6*n^2 + 6*n -1))/9.
%F A143978 a(n) = (A102283(n+1) - A102283(n) + A103115(n+1))/9
%F A143978 a(n) = (A131713(n) + A103115(n+1))/9. (End)
%F A143978 Sum_{n>=1} 1/a(n) = 3/2 + (tan(Pi/(2*sqrt(3)))-1)*Pi/(2*sqrt(3)). - _Amiram Eldar_, Sep 27 2022
%F A143978 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
%p A143978 A143978:= n-> (6*n*(n+1) -1 + `mod`(n+2,3) - `mod`(n+1,3))/9;
%p A143978 seq(A143978(n), n=1..60); # _G. C. Greubel_, May 27 2020
%t A143978 Table[(6*n^2 +6*n -1  + Mod[n+2, 3] - Mod[n+1, 3])/9, {n, 60}] (* _G. C. Greubel_, May 27 2020 *)
%t A143978 Table[Floor[2n (n+1)/3],{n,60}] (* or *) LinearRecurrence[{2,-1,1,-2,1},{1,4,8,13,20},60] (* _Harvey P. Dale_, Aug 12 2025 *)
%Y A143978 Cf. A000217, A030511, A042965 (first differences), A106510, A131534, A143979.
%Y A143978 Cf. A049347, A102283, A103115, A131713.
%K A143978 nonn,easy
%O A143978 1,2
%A A143978 _Clark Kimberling_, Sep 06 2008