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A347902 a(n) = a(n-1) + a(n-3) + a(n-4) with initial values a(0) = 8, a(1)=5, a(2) = 13, a(3) = 30.

%I A347902 #42 Mar 19 2024 21:22:41
%S A347902 8,5,13,30,43,61,104,177,281,446,727,1185,1912,3085,4997,8094,13091,
%T A347902 21173,34264,55449,89713,145150,234863,380025,614888,994901,1609789,
%U A347902 2604702,4214491,6819181,11033672,17852865,28886537,46739390,75625927,122365329
%N A347902 a(n) = a(n-1) + a(n-3) + a(n-4) with initial values a(0) = 8, a(1)=5, a(2) = 13, a(3) = 30.
%C A347902 For n >= 3, a(n) is also the number of ways to tile this "central staircase" figure of length n with squares and dominoes; this is the picture for length n=10:
%C A347902              _
%C A347902            _|_|_
%C A347902    _______|_|_|_|_____
%C A347902   |_|_|_|_|_|_|_|_|_|_|
%H A347902 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,1).
%F A347902 G.f.: (8 - 3*x + 8*x^2 + 9*x^3)/((1-x-x^2)*(1+x^2)).
%F A347902 a(n) = (7*Lucas(n+3) + 6*i^(n*(n+1))*(3-(-1)^n))/5 where i = sqrt(-1).
%F A347902 E.g.f.: (12*cos(x) - 24*sin(x) + 14*exp(x/2)*(2*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)))/5. - _Stefano Spezia_, Sep 18 2021
%F A347902 5*a(n) = 7*A000032(n+3) - 12 *(-1)^floor((n-1)/2)*A000034(n). - _R. J. Mathar_, Sep 30 2021
%F A347902 From _Greg Dresden_, Mar 19 2024: (Start)
%F A347902 a(2*n) = (7*Lucas(2*n+3) + 12*(-1)^n)/5.
%F A347902 a(2*n+1) = (7*Lucas(2*n+4) - 24*(-1)^n)/5. (End)
%e A347902 Here is one of the a(10)=727 tilings for n=10.
%e A347902              _
%e A347902            _| |_
%e A347902    _______| |_|_|_____
%e A347902   |_|___|_|_|___|_|___|
%t A347902 LinearRecurrence[{1, 0, 1, 1}, {8, 5, 13, 30}, 33]
%Y A347902 Cf. A000032, A000034.
%K A347902 nonn,easy
%O A347902 0,1
%A A347902 _Reeva Bohra_ and _Greg Dresden_, Sep 18 2021