A347902 - OEIS

%I #42 Mar 19 2024 21:22:41

%S 8,5,13,30,43,61,104,177,281,446,727,1185,1912,3085,4997,8094,13091,

%T 21173,34264,55449,89713,145150,234863,380025,614888,994901,1609789,

%U 2604702,4214491,6819181,11033672,17852865,28886537,46739390,75625927,122365329

%N 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 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              _

%C            _|_|_

%C    _______|_|_|_|_____

%C   |_|_|_|_|_|_|_|_|_|_|

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,1).

%F G.f.: (8 - 3*x + 8*x^2 + 9*x^3)/((1-x-x^2)*(1+x^2)).

%F a(n) = (7*Lucas(n+3) + 6*i^(n*(n+1))*(3-(-1)^n))/5 where i = sqrt(-1).

%F 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 5*a(n) = 7*A000032(n+3) - 12 *(-1)^floor((n-1)/2)*A000034(n). - _R. J. Mathar_, Sep 30 2021

%F From _Greg Dresden_, Mar 19 2024: (Start)

%F a(2*n) = (7*Lucas(2*n+3) + 12*(-1)^n)/5.

%F a(2*n+1) = (7*Lucas(2*n+4) - 24*(-1)^n)/5. (End)

%e Here is one of the a(10)=727 tilings for n=10.

%e              _

%e            _| |_

%e    _______| |_|_|_____

%e   |_|___|_|_|___|_|___|

%t LinearRecurrence[{1, 0, 1, 1}, {8, 5, 13, 30}, 33]

%Y Cf. A000032, A000034.

%K nonn,easy

%O 0,1

%A _Reeva Bohra_ and _Greg Dresden_, Sep 18 2021