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%I #25 Feb 23 2024 07:28:20
%S 1,0,3,4,9,12,19,24,33,40,51,60,73,84,99,112,129,144,163,180,201,220,
%T 243,264,289,312,339,364,393,420,451,480,513,544,579,612,649,684,723,
%U 760,801,840,883,924,969,1012,1059,1104,1153,1200,1251,1300,1353,1404
%N Expansion of (1-2*x+3*x^2)/((1+x)*(1-x)^3).
%C Partial sums of A097062. Pairwise sums are A002061. Binomial transform is essentially A007466.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F G.f. : (1-2*x+3*x^2)/((1-x^2)(1-x)^2).
%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
%F a(n) = Sum_{k=0..n} (k^2-k+1)*(-1)^(n-k).
%F a(2n) = A058331(n); a(2n+1) = A046092(n). - _R. J. Mathar_, Oct 27 2008
%F a(n) = binomial(n+1, 2) - ceiling((n+1)/2) + 2((n+1) mod 2). - _Wesley Ivan Hurt_, Mar 08 2014
%F a(n) = 2*floor(n/2) + ceiling((n-1)^2/2). - _M. Ryan Julian Jr._, Sep 10 2019
%F a(n) = A326296(n + 1, n) for n > 0. - _Andrew Howroyd_, Sep 23 2019
%p A097063:=n->(1/4) + (3/4)*(-1)^n + (1/2)*n^2; seq(A097063(n), n=0..50); # _Wesley Ivan Hurt_, Mar 08 2014
%t Table[(1/4) + (3/4)*(-1)^n + (1/2)*n^2, {n, 0, 50}] (* _Wesley Ivan Hurt_, Mar 08 2014 *)
%Y A diagonal of A326296.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Jul 22 2004
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