A269403 Expansion of x*(2 - x + 2*x^2 + x^3)/((1 - x)^3*(1 + x + x^2 + x^3)).

0, 2, 3, 6, 10, 16, 21, 28, 36, 46, 55, 66, 78, 92, 105, 120, 136, 154, 171, 190, 210, 232, 253, 276, 300, 326, 351, 378, 406, 436, 465, 496, 528, 562, 595, 630, 666, 704, 741, 780, 820, 862, 903, 946, 990, 1036, 1081, 1128, 1176, 1226, 1275, 1326, 1378, 1432, 1485

Offset

0,2

Comments

Partial sums of A080412.

Links

Table of n, a(n) for n=0..54.

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Formula

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

a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6).

a(n) = (2*n^2 + 2*n + 2*sin((Pi*n)/2) - (-1)^n + 1)/4.

Sum_{n>=1} 1/a(n) = 1.495144413654306177...

Example

a(0) = 0;
a(1) = 0 + 2 = 2;
a(2) = 0 + 2 + 1 = 3;
a(3) = 0 + 2 + 1 + 3 = 6;
a(4) = 0 + 2 + 1 + 3 + 4 = 10;
a(5) = 0 + 2 + 1 + 3 + 4 + 6 = 16;
a(6) = 0 + 2 + 1 + 3 + 4 + 6 + 5 = 21;
a(7) = 0 + 2 + 1 + 3 + 4 + 6 + 5 + 7 = 28;
a(8) = 0 + 2 + 1 + 3 + 4 + 6 + 5 + 7 + 8 = 36;
a(9) = 0 + 2 + 1 + 3 + 4 + 6 + 5 + 7 + 8 + 10 = 46, etc.

Mathematica

LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 2, 3, 6, 10, 16}, 55]
Table[(2 n^2 + 2 n + 2 Sin[(Pi n)/2] - (-1)^n + 1)/4, {n, 0, 54}]

Crossrefs

Cf. A001477, A080412, A116996.

Sequence in context: A051891 A108062 A347117 * A075623 A024801 A324742

Adjacent sequences: A269400 A269401 A269402 * A269404 A269405 A269406

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Feb 25 2016

Status

approved