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%I #25 Jan 01 2024 11:52:14
%S 1,3,15,69,321,1491,6927,32181,149505,694563,3226767,14990757,
%T 69643329,323545587,1503112335,6983086101,32441681409,150715983939,
%U 700188979983,3252903871749,15112182426945,70207441323027,326166312572943
%N a(0)=1, a(1)=3, a(n) = 4*a(n-1) + 3*a(n-2) for n > 1.
%C a(n) is the number of compositions of n when there are 3 types of 1 and 6 types of other natural numbers. - _Milan Janjic_, Aug 13 2010
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,3).
%F G.f.: (1-x)/(1-4*x-3*x^2).
%F a(n) = Sum_{k=0..n} 3^k*A122542(n,k).
%F Limit_{n->infinity} a(n+1)/a(n) = 2 + sqrt(7) = 4.645751311064....
%F a(n) = ((7+sqrt(7))/14)*(2+sqrt(7))^n + ((7-sqrt(7))/14)*(2-sqrt(7))^n. - _Richard Choulet_, Nov 20 2008
%t LinearRecurrence[{4,3},{1,3},30] (* _Harvey P. Dale_, Mar 18 2023 *)
%o (PARI) Vec((1-x)/(1-4*x-3*x^2) + O(x^30)) \\ _Michel Marcus_, Feb 04 2022
%Y Cf. A122542.
%K nonn
%O 0,2
%A _Philippe Deléham_, Sep 20 2006, Sep 22 2006
%E Corrected by _T. D. Noe_, Nov 07 2006
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