A374729 - OEIS

%I #11 Jul 19 2024 14:39:54

%S 0,1,2,4,7,12,21,40,76,139,254,466,855,1576,2905,5340,9816,18053,

%T 33202,61076,112351,206636,380045,699012,1285684,2364759,4349502,

%U 7999954,14714159,27063568,49777681,91555464,168396816,309729961,569682082,1047808756

%N Number of tilings using squares, dominos, and flexible trominos of a strip of length n-1 and with an n-th cell placed on top of the middle of the strip.

%C As an illustration, here are the figures for n=8 and n=9, respectively.

%C        _                    _

%C  _____|_|_____      _______|_|_____

%C |_|_|_|_|_|_|_|,   |_|_|_|_|_|_|_|_|.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,0,2,2,0,-1,-1).

%F a(n) = a(n-1) + 2*a(n-3) + 2*a(n-5) + 2*a(n-6) - a(n-8) - a(n-9).

%F a(2*n) = a(2*n-1) + a(2*n-3) + a(2*n-4) + 3*a(2*n-5) + 2*a(2*n-6) + a(2*n-7).

%F a(2*n) = A000073(2*n+1) + A000073(n+1)*(A000073(n+1) + 2*A000073(n)).

%F a(2*n+1) = a(2*n) + a(2*n-1) + a(2*n-3) + a(2*n-4) + a(2*n-5).

%F a(2*n+1) = A000073(2*n+2) + A000073(n+1)^2 + A000073(n+2)*(A000073(n+1) + A000073(n)).

%F G.f.: x*(1 + x + 2*x^2 + x^3 + x^4 - x^5 - x^6)/(1 - x - 2*x^3 - 2*x^5 -

%F  2*x^6 + x^8 + x^9).

%e For n=8, here is one of a(8)=76 possible tilings with squares, dominos, and flexible trominos.

%e        _

%e  _____| |_____

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

%t LinearRecurrence[{1, 0, 2, 0, 2, 2, 0, -1, -1}, {0, 1, 2, 4, 7, 12, 21, 40, 76}, 40]

%Y Cf. A000073.

%K nonn,easy

%O 0,3

%A _Greg Dresden_ and Yinuo Zhu, Jul 17 2024