A269098 - OEIS

%I #21 Sep 08 2022 08:46:15

%S 1,2,3,3,4,7,5,6,11,7,8,15,9,10,19,11,12,23,13,14,27,15,16,31,17,18,

%T 35,19,20,39,21,22,43,23,24,47,25,26,51,27,28,55,29,30,59,31,32,63,33,

%U 34,67,35,36,71,37,38,75,39,40,79,41,42,83,43,44,87,45

%N Expansion of (1 + 2*x + 3*x^2 + x^3 + x^5)/(1 - x^3)^2.

%H Ilya Gutkovskiy, <a href="/A269098/a269098.pdf">Expanded graphical example</a>

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

%F G.f.: (1 + 2*x + 3*x^2 + x^3 + x^5)/(1 - x^3)^2.

%F a(3*n) + a(3*n+1) = a(3*n+2).

%F a(n) = (8*(n+1) + (1-2*n)*cos(2*n*Pi/3) + sqrt(3)*(3-2*n)*sin(2*n*Pi/3))/9.

%F a(n) = 1 + n + (1-2*n)*floor(n/3)/3 + 2*(n-2)*floor((n+1)/3)/3. - _Vaclav Kotesovec_, Feb 25 2016

%e a(0) = 1;

%e a(1) = 2;

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

%e a(3) = 3;

%e a(4) = 4;

%e a(5) = 3 + 4 = 7,

%e a(6) = 5;

%e a(7) = 6;

%e a(8) = 5 + 6 = 11, etc.

%t LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 2, 3, 3, 4, 7}, 78]

%t CoefficientList[Series[(1 + 2 x + 3 x^2 + x^3 + x^5)/(1 - x^3)^2, {x, 0, 77}], x]

%t Table[(8 (n + 1) + (1 - 2 n) Cos[2 n (Pi/3)] + Sqrt[3] (3 - 2 n) Sin[2 n (Pi/3)])/9, {n, 0, 77}]

%o (PARI) Vec((1 + 2*x + 3*x^2 + x^3 + x^5)/(1 - x^3)^2 + O(x^80)) \\ _Michel Marcus_, Feb 20 2016

%o (Magma) I:=[1,2,3,3,4,7]; [n le 6 select I[n] else 2*Self(n-3)-Self(n-6): n in [1..70]]; // _Vincenzo Librandi_, Feb 20 2016

%Y Cf. A000027, A004767.

%K nonn,easy

%O 0,2

%A _Ilya Gutkovskiy_, Feb 19 2016