A099623 - OEIS

%I #14 Jan 15 2025 06:31:42

%S 0,0,1,6,27,104,369,1242,4039,12828,40077,123758,379011,1153872,

%T 3498025,10572354,31884543,96010436,288788613,867967830,2607282235,

%U 7828953720,23501774241,70536546986,211674885687,635160738924,1905765565309

%N a(n) = Sum_{k=0..floor(n/2)} C(n-k,k+2)*2^(n-k-2)*(3/2)^k.

%C In general a(n) = Sum_{k=0..floor(n/2)} C(n-k,k+2)*u^(n-k-2)*(v/u)^k has g.f. x^2/((1-u*x)^2*(1-u*x-v*x^2)) and satisfies the recurrence a(n) = 3*u*a(n-1)-(3*u^2-v)*a(n-2)+(u^3-2*u*v)*a(n-3)+u^2*v*a(n-4).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,-4,12).

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

%F a(n) = 6*a(n-1)-9*a(n-2)-4*a(n-3)+12*a(n-4).

%F a(n) = -(n/3+7/9)*2^n +(-1)^n/36 +3^(n+1)/4. - _R. J. Mathar_, Dec 16 2024

%t LinearRecurrence[{6,-9,-4,12},{0,0,1,6},30] (* _Harvey P. Dale_, Mar 27 2016 *)

%K easy,nonn

%O 0,4

%A _Paul Barry_, Oct 25 2004