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%I #32 Sep 08 2022 08:45:11
%S 1,2,6,12,23,38,60,88,125,170,226,292,371,462,568,688,825,978,1150,
%T 1340,1551,1782,2036,2312,2613,2938,3290,3668,4075,4510,4976,5472,
%U 6001,6562,7158,7788,8455,9158,9900,10680,11501,12362,13266,14212,15203,16238
%N Partial sums of A084263.
%C Partial sums give A084569.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F a(n) = (-1)^n/4 + (2n^3 + 6n^2 + 10n + 9)/12.
%F a(n) = Sum_{j=0..n} (Sum_{i=0..j} (i + (-1)^i)).
%F From _Arun Giridhar_, Apr 03 2015: (Start)
%F a(n) = ceiling(A006527(n+1) / 2).
%F a(n) = ceiling((n^3 + 3n^2 + 5n + 3)/6).
%F (End)
%F G.f.: (1-x+2*x^2)/((1+x)*(1-x)^4). - _Vincenzo Librandi_, Apr 04 2015
%t LinearRecurrence[{3,-2,-2,3,-1},{1,2,6,12,23},50] (* _Harvey P. Dale_, Nov 12 2014 *)
%t CoefficientList[Series[(1 - x + 2 x^2) / ((1 + x) (1 - x)^4), {x, 0, 50}], x] (* _Vincenzo Librandi_, Apr 04 2015 *)
%o (PARI) a(n) = sum(j=0, n, sum(i=0, j, (i+(-1)^i)));
%o vector(50, n, n--; a(n)) \\ _Michel Marcus_, Apr 04 2015
%o (Magma) [(-1)^n/4 + (2*n^3+6*n^2+10*n+ 9)/12: n in [0..50]]; // _Vincenzo Librandi_, Apr 04 2015
%K easy,nonn
%O 0,2
%A _Paul Barry_, May 31 2003
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