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A385933 Number of ways to tile a "central bump" strip of length n with 1 X 1 squares and 1 X 3 rectangles.

%I A385933 #9 Jul 17 2025 16:10:58
%S A385933 4,9,13,25,30,35,52,78,121,189,271,388,561,812,1204,1785,2617,3837,
%T A385933 5602,8179,12000,17606,25825,37881,55483,81264,119089,174520,255828,
%U A385933 375017,549589,805425,1180342,1729779,2535196,3715630,5445561,7980917,11696455,17141772
%N A385933 Number of ways to tile a "central bump" strip of length n with 1 X 1 squares and 1 X 3 rectangles.
%C A385933 a(n) is the number of ways to tile this "central bump" strip of length n (shown here at n=18) with 1 X 1 squares and 1 X 3 rectangles which can be horizontal or vertical:
%C A385933                    _
%C A385933                  _|_|_
%C A385933  _______________|_|_|_|_____________
%C A385933 |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
%C A385933                 |_|_|_|
%C A385933                   |_|
%F A385933 a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) + a(n-6) - a(n-9).
%F A385933 a(2*n) = a(2*n-2) + a(2*n-4) + 2*a(2*n-6) + a(2*n-7) + a(2*n-8).
%F A385933 a(2*n+1) = a(2*n-1) + 2*a(2*n-4) + a(2*n-5) + 2*a(2*n-6).
%F A385933 a(2*n+3) = 25*b(n)^2 + 26*b(n)*b(n-2) + 10*b(n)*b(n-1) + 9*b(n-2)^2 + 8*b(n-1)*b(n-2) for b(n) = A000930(n) the Narayana Cow sequence.
%F A385933 G.f.: (4 + 5*x + 4*x^2 + 8*x^3 - 3*x^5 - 8*x^6 - x^7)/(1 - x - x^3 + x^4 - x^5 - x^6 + x^9).
%e A385933 For n = 0 there is no horizontal strip but there is still the "central bump". Here are the a(n) = 4 ways to tile this (disjoint) structure with 1 X 1 squares and 1 X 3 rectangles.
%e A385933    _           _           _           _
%e A385933  _|_|_       _|_|_       _|_|_       _|_|_
%e A385933 |_|_|_|     |_____|     |_|_|_|     |_____|
%e A385933  _____       _____       _____       _____
%e A385933 |_|_|_|     |_|_|_|     |_____|     |_____|
%e A385933   |_|         |_|         |_|         |_|
%t A385933 LinearRecurrence[{1,0,1,-1,1,1,0,0,-1},{4,9,13,25,30,35,52,78,121},61]
%Y A385933 Cf. A000930.
%K A385933 nonn
%O A385933 0,1
%A A385933 _Greg Dresden_ and Saim Usmani, Jul 12 2025