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A082667 a(n) = floor(2n/3) * ceiling(2n/3) / 2.

%I A082667 #40 May 11 2025 09:26:11
%S A082667 0,1,2,3,6,8,10,15,18,21,28,32,36,45,50,55,66,72,78,91,98,105,120,128,
%T A082667 136,153,162,171,190,200,210,231,242,253,276,288,300,325,338,351,378,
%U A082667 392,406,435,450,465,496,512,528,561,578,595,630,648,666,703,722,741
%N A082667 a(n) = floor(2n/3) * ceiling(2n/3) / 2.
%C A082667 Prefixing with 0,0,0 gives the sequence c(n) defined as the number of (x,y,z) satisfying 2w = 3x-3y where w,x,y are all in {1,...,n}, for n>=0; see the Formula section.
%C A082667 For n >= 2, numbers k such that floor(sqrt(2k)+1/2) | 2k. - _Wesley Ivan Hurt_, Dec 01 2020
%H A082667 Amiram Eldar, <a href="/A082667/b082667.txt">Table of n, a(n) for n = 1..10000</a>
%H A082667 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F A082667 a(n) = a(n-1) + 2a(n-3) - 2a(n-4) - a(n-6) + a(n-7), (with 0,0,0 prefixed as in the Comments section). - _Clark Kimberling_, Apr 15 2012
%F A082667 a(n) = floor((n + 1)/3)*(n - floor((n + 1)/3)). - _Wesley Ivan Hurt_, Jun 06 2014
%F A082667 G.f.: -x^2*(1+x)*(1+x^2) / ( (1+x+x^2)^2*(x-1)^3 ). - _R. J. Mathar_, Jun 07 2014
%F A082667 From _Amiram Eldar_, May 10 2025: (Start)
%F A082667 Sum_{n>=2} 1/a(n) = 2 + Pi^2/12.
%F A082667 Sum_{n>=2} (-1)^n/a(n) = Pi - Pi^2/24 - 2. (End)
%F A082667 E.g.f.: exp(-x/2)*(2*exp(3*x/2)*(3*x^2 + 3*x - 1) - (3*x - 2)*cos(sqrt(3)*x/2) + sqrt(3)*x*sin(sqrt(3)*x/2))/27. - _Stefano Spezia_, May 11 2025
%t A082667 n2[n_]:=Module[{c=2*n/3},(Floor[c]Ceiling[c])/2]; Array[n2,60] (* _Harvey P. Dale_, Feb 03 2012 *)
%t A082667 LinearRecurrence[{1,0,2,-2,0,-1,1},{0,1,2,3,6,8,10},60] (* _Robert G. Wilson v_, Jun 06 2014 *)
%o A082667 (PARI) a(n) = (2*n\3) * ceil(2*n/3) / 2; \\ _Amiram Eldar_, May 10 2025
%Y A082667 Cf. A008130, A151842 (first differences).
%K A082667 nonn,easy
%O A082667 1,3
%A A082667 _Reinhard Zumkeller_, May 18 2003