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%I #51 Sep 08 2022 08:46:12
%S 0,0,1,3,6,9,14,18,25,30,39,45,56,63,76,84,99,108,125,135,154,165,186,
%T 198,221,234,259,273,300,315,344,360,391,408,441,459,494,513,550,570,
%U 609,630,671,693,736,759,804,828,875,900,949,975,1026,1053,1106,1134
%N Start with all terms set to 0. Then add n to the next n+2 terms for n=0,1,2,... .
%H Reinhard Zumkeller, <a href="/A258087/b258087.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F a(n) = (6*n^2+2*n-11+(2*n-5)*(-1)^n)/16+0^n.
%F a(n) = Sum_{i=1..n-1} (3*i+2)/4+(2-i)*(-1)^i/4.
%F From _Robert Israel_, May 19 2015: (Start)
%F G.f.: x^2*(x^3-x^2-2*x-1)/((x+1)^2*(x-1)^3).
%F E.g.f.: 1 + exp(x)*(6*x^2+8*x-11)/16 - exp(-x)*(2*x+5)/16.
%F a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n >= 6. (End)
%F From _Bruno Berselli_, May 20 2015: (Start)
%F a(n) = a(-n) for n odd, a(n) = a(-n)+n/2 otherwise.
%F a(n) = (floor(n/2)+1)*(floor(n/2)+2*floor((n-1)/2))/2 for n>0. Therefore, after 3, all terms of the sequence are composite. (End)
%F a(n) = Sum_{i=floor((n-1)/2)..n-1} i, for n>0. - _Wesley Ivan Hurt_, Apr 11 2016
%e n | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10, ...
%e __________________________________________
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e + 0, 0
%e + 1, 1, 1
%e + 2, 2, 2, 2
%e + 3, 3, 3, 3, 3
%e + 4, 4, 4, 4, 4, 4
%e + 5, 5, 5, 5, 5, 5, 5
%e + 6, 6, 6, 6, 6, 6, 6, 6
%e + 7, 7, 7, 7, 7, 7, 7, 7, 7, 7
%e + 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8
%e + 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9
%e + ...
%e __________________________________________
%e a(n)|0, 0, 1, 3, 6, 9,14,18,25,30,39, ...
%p A258087:=n->(6*n^2+2*n-11+(2*n-5)*(-1)^n)/16+0^n: seq(A258087(n), n=0..100);
%t Join[{0}, Table[(6 n^2 + 2 n - 11 + (2 n - 5) (-1)^n)/16, {n, 100}]]
%t Table[Total@ Range[Floor[(n - 1)/2], n - 1], {n, 55}] (* _Michael De Vlieger_, Apr 11 2016 *)
%o (Magma) [(6*n^2+2*n-11+(2*n-5)*(-1)^n)/16+0^n: n in [0..60]]; // _Vincenzo Librandi_, May 20 2015
%o (Sage) [(6*n^2+2*n-11+(2*n-5)*(-1)^n)/16+0^n for n in (0..60)] # _Bruno Berselli_, May 20 2015
%o (Haskell)
%o a258087 n = a258087_list !! n
%o a258087_list = f 0 [0] $
%o map (\i -> take (i + 1) (repeat 0) ++ replicate (i + 2) i) [0..] where
%o f i ys@(y:_) (xs:xss) = (ys !! i) :
%o f (i + 1) (zipWith (+) (ys ++ repeat 0) xs) xss
%o -- _Reinhard Zumkeller_, May 21 2015
%o (PARI) a(n) = if (n==0, 0, sum(k = (n-1)\2, n-1, k)); \\ _Michel Marcus_, Apr 11 2016
%o (PARI) x='x+O('x^99); concat([0, 0], Vec(x^2*(x^3-x^2-2*x-1)/((x+1)^2*(x-1)^3))) \\ _Altug Alkan_, Apr 11 2016
%Y Cf. A272058.
%K nonn,easy
%O 0,4
%A _Wesley Ivan Hurt_, May 19 2015
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