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%I #31 Oct 24 2022 02:24:22
%S 1,3,6,9,13,18,23,29,36,43,51,60,69,79,90,101,113,126,139,153,168,183,
%T 199,216,233,251,270,289,309,330,351,373,396,419,443,468,493,519,546,
%U 573,601,630,659,689,720,751,783,816,849,883,918,953,989,1026,1063,1101
%N a(n) = floor(n*(n+3)/3).
%C Fourth diagonal of A143974, associated with counting unit squares in a lattice.
%H Vincenzo Librandi, <a href="/A143975/b143975.txt">Table of n, a(n) for n = 1..3000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F a(n) = floor(n*(n+3)/3).
%F From _R. J. Mathar_, Oct 05 2009: (Start)
%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
%F G.f.: x*(-1 - x - x^2 + x^3)/( (1 + x + x^2) * (x-1)^3). (End)
%F 9*a(n) = 3*n^2 + 9*n - 2 + A099837(n+3). - _R. J. Mathar_, Apr 26 2022
%F Sum_{n>=1} 1/a(n) = 4/3 + (tan((sqrt(13)+2)*Pi/6) - cot((sqrt(13)+1)*Pi/6)) * Pi/sqrt(13). - _Amiram Eldar_, Oct 01 2022
%F E.g.f.: (exp(x)*(3*x*(4 + x) - 2) + 2*exp(-x/2)*cos(sqrt(3)*x/2))/9. - _Stefano Spezia_, Oct 24 2022
%e Main diagonal of A143974: (0,1,3,5,8,12,...) = A000212;
%e 2nd diagonal: (0,2,4,6,10,14,18,...) = A128422;
%e 3rd diagonal: (1,2,5,8,11,16,21,...) = A032765;
%e 4th diagonal: (1,3,6,9,13,18,23,...) = A143975.
%t a[n_] := Floor[n*(n+3)/3]; Array[a, 60] (* _Amiram Eldar_, Oct 01 2022 *)
%o (Magma) [Floor(n*(n+3)/3): n in [1..60]]; // _Vincenzo Librandi_, May 08 2011
%Y Cf. A099837, A143974.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Sep 06 2008
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