A258087 Start with all terms set to 0. Then add n to the next n+2 terms for n=0,1,2,... .

0, 0, 1, 3, 6, 9, 14, 18, 25, 30, 39, 45, 56, 63, 76, 84, 99, 108, 125, 135, 154, 165, 186, 198, 221, 234, 259, 273, 300, 315, 344, 360, 391, 408, 441, 459, 494, 513, 550, 570, 609, 630, 671, 693, 736, 759, 804, 828, 875, 900, 949, 975, 1026, 1053, 1106, 1134

Offset

0,4

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(n) = (6*n^2+2*n-11+(2*n-5)*(-1)^n)/16+0^n.

a(n) = Sum_{i=1..n-1} (3*i+2)/4+(2-i)*(-1)^i/4.

From Robert Israel, May 19 2015: (Start)

G.f.: x^2*(x^3-x^2-2*x-1)/((x+1)^2*(x-1)^3).

E.g.f.: 1 + exp(x)*(6*x^2+8*x-11)/16 - exp(-x)*(2*x+5)/16.

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n >= 6. (End)

From Bruno Berselli, May 20 2015: (Start)

a(n) = a(-n) for n odd, a(n) = a(-n)+n/2 otherwise.

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)

a(n) = Sum_{i=floor((n-1)/2)..n-1} i, for n>0. - Wesley Ivan Hurt, Apr 11 2016

Example

n  | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10, ...
__________________________________________
     0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
+       0, 0
+          1, 1, 1
+             2, 2, 2, 2
+                3, 3, 3, 3, 3
+                   4, 4, 4, 4, 4, 4
+                      5, 5, 5, 5, 5, 5, 5
+                         6, 6, 6, 6, 6, 6, 6, 6
+                            7, 7, 7, 7, 7, 7, 7, 7, 7, 7
+                               8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8
+                                  9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9
+                                      ...
__________________________________________
a(n)|0, 0, 1, 3, 6, 9,14,18,25,30,39,  ...

Maple

A258087:=n->(6*n^2+2*n-11+(2*n-5)*(-1)^n)/16+0^n: seq(A258087(n), n=0..100);

Mathematica

Join[{0}, Table[(6 n^2 + 2 n - 11 + (2 n - 5) (-1)^n)/16, {n, 100}]]
Table[Total@ Range[Floor[(n - 1)/2], n - 1], {n, 55}] (* Michael De Vlieger, Apr 11 2016 *)

Prog

(Magma) [(6*n^2+2*n-11+(2*n-5)*(-1)^n)/16+0^n: n in [0..60]]; // Vincenzo Librandi, May 20 2015
(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
(Haskell)
a258087 n = a258087_list !! n
a258087_list = f 0 [0] $
   map (\i -> take (i + 1) (repeat 0) ++ replicate (i + 2) i) [0..] where
   f i ys@(y:_) (xs:xss) = (ys !! i) :
                           f (i + 1) (zipWith (+) (ys ++ repeat 0) xs) xss
-- Reinhard Zumkeller, May 21 2015
(PARI) a(n) = if (n==0, 0, sum(k = (n-1)\2, n-1, k)); \\ Michel Marcus, Apr 11 2016
(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

Crossrefs

Cf. A272058.

Sequence in context: A000791 A027424 A294476 * A191130 A319738 A310165

Adjacent sequences: A258084 A258085 A258086 * A258088 A258089 A258090

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 19 2015

Status

approved