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A385717 a(n) = a(n-1) + a(n-2) + a(n-3), with a(1) = 4, a(2) = 13, a(3) = 42.

%I A385717 #12 Jul 27 2025 00:21:05
%S A385717 4,13,42,59,114,215,388,717,1320,2425,4462,8207,15094,27763,51064,
%T A385717 93921,172748,317733,584402,1074883,1977018,3636303,6688204,12301525,
%U A385717 22626032,41615761,76543318,140785111,258944190,476272619
%N A385717 a(n) = a(n-1) + a(n-2) + a(n-3), with a(1) = 4, a(2) = 13, a(3) = 42.
%C A385717 For n >= 3, a(n) is the number of ways to tile this shape of length n with 1 X 1 squares, 1 X 2 dominos, and 1 X 3 trominos:
%C A385717    ___
%C A385717  _|_|_|_________
%C A385717 |_|_|_|_|_|_|_|_|
%C A385717   |_|_|
%C A385717 As an example, here is one of the a(8) = 717 ways to tile this shape of length 8:
%C A385717    ___
%C A385717  _| |_|_________
%C A385717 |_| | |_|_|_____|
%C A385717   |_|_|
%H A385717 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1).
%F A385717 a(n) = 9*A354080(n-2) + 2*A100683(n) for n >= 2.
%F A385717 G.f.: x*(4 + 9*x + 25*x^2)/(1 - x - x^2 - x^3). - _Stefano Spezia_, Jul 08 2025
%t A385717 LinearRecurrence[{1, 1, 1}, {4, 13, 42}, 30]
%Y A385717 Cf. A100683, A354080.
%K A385717 nonn,easy
%O A385717 1,1
%A A385717 _Greg Dresden_ and Jiarui Zhou, Jul 07 2025