A276226 - OEIS

%I #16 Sep 08 2022 08:46:17

%S 0,6,8,22,58,146,372,948,2414,6148,15658,39878,101562,258660,658760,

%T 1677742,4272904,10882310,27715266,70585746,179769068,457839148,

%U 1166033110,2969674436,7563221130,19262149806,49057195178,124939761292,318198867568,810394691606,2063928012072,5256449583318,13387221870314,34094821336018

%N a(n+3) = 2*a(n+2) + a(n+1) + a(n) with a(0)=0, a(1)=6, a(2)=8.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,1).

%F Let p = (4*(61 + 9*sqrt(29)))^(1/3), q = (4*(61 - 9*sqrt(29)))^(1/3), and x = (1/6)*(4 + p + q) then x^n = (1/6)*(2*A276225(n) + a(n)*(p + q) + A077939(n-1)*(p^2 + q^2)).G.f.: 2*(3*x - 2*x^2)/(1 - 2*x - x^2 - x^3).

%t LinearRecurrence[{2, 1, 1}, {0, 6, 8}, 50]

%t CoefficientList[Series[2 (3 x - 2 x^2)/(1 - 2 x - x^2 - x^3), {x, 0, 33}], x] (* _Michael De Vlieger_, Aug 25 2016 *)

%o (Magma) I:=[0,6,8]; [n le 3 select I[n] else 2*Self(n-1)+ Self(n-2)+Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Aug 25 2016

%o (PARI) concat(0, Vec(2*(3*x-2*x^2)/(1-2*x-x^2-x^3) + O(x^99))) \\ _Altug Alkan_, Aug 25 2016

%Y Cf. A276225, A077939.

%K nonn,easy

%O 0,2

%A _G. C. Greubel_, Aug 24 2016