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Expansion of x*(2 - x + 2*x^2 + x^3)/((1 - x)^3*(1 + x + x^2 + x^3)).
0

%I #9 Feb 25 2016 17:09:39

%S 0,2,3,6,10,16,21,28,36,46,55,66,78,92,105,120,136,154,171,190,210,

%T 232,253,276,300,326,351,378,406,436,465,496,528,562,595,630,666,704,

%U 741,780,820,862,903,946,990,1036,1081,1128,1176,1226,1275,1326,1378,1432,1485

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

%C Partial sums of A080412.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).

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

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

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

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

%e a(0) = 0;

%e a(1) = 0 + 2 = 2;

%e a(2) = 0 + 2 + 1 = 3;

%e a(3) = 0 + 2 + 1 + 3 = 6;

%e a(4) = 0 + 2 + 1 + 3 + 4 = 10;

%e a(5) = 0 + 2 + 1 + 3 + 4 + 6 = 16;

%e a(6) = 0 + 2 + 1 + 3 + 4 + 6 + 5 = 21;

%e a(7) = 0 + 2 + 1 + 3 + 4 + 6 + 5 + 7 = 28;

%e a(8) = 0 + 2 + 1 + 3 + 4 + 6 + 5 + 7 + 8 = 36;

%e a(9) = 0 + 2 + 1 + 3 + 4 + 6 + 5 + 7 + 8 + 10 = 46, etc.

%t LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 2, 3, 6, 10, 16}, 55]

%t Table[(2 n^2 + 2 n + 2 Sin[(Pi n)/2] - (-1)^n + 1)/4, {n, 0, 54}]

%Y Cf. A001477, A080412, A116996.

%K nonn,easy

%O 0,2

%A _Ilya Gutkovskiy_, Feb 25 2016