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A347493 a(0) = 1, a(1) = 0, a(2) = a(3) = 1; thereafter, a(n) = a(n-1) + a(n-2) + a(n-4).

%I A347493 #38 Aug 17 2023 20:50:43
%S A347493 1,0,1,1,3,4,8,13,24,41,73,127,224,392,689,1208,2121,3721,6531,11460,
%T A347493 20112,35293,61936,108689,190737,334719,587392,1030800,1808929,
%U A347493 3174448,5570769,9776017,17155715,30106180,52832664,92714861,162703240,285524281,501060185,879299327,1543062752
%N A347493 a(0) = 1, a(1) = 0, a(2) = a(3) = 1; thereafter, a(n) = a(n-1) + a(n-2) + a(n-4).
%C A347493 a(n) is also the number of ways to tile a strip of length n with squares, dominoes, and tetrominoes such that the first tile is NOT a square. As such, it completes the set of such tilings with A005251 (first tile is NOT a domino), A005314 (first tile is NOT a tetromino), and A060945 (no restrictions on first tile).
%H A347493 Michael De Vlieger, <a href="/A347493/b347493.txt">Table of n, a(n) for n = 0..4096</a>
%H A347493 Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, <a href="https://ajc.maths.uq.edu.au/pdf/84/ajc_v84_p398.pdf">Dyck Paths with catastrophes modulo the positions of a given pattern</a>, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.
%H A347493 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,1).
%F A347493 a(n) = 2*A060945(n) - A005251(n) - A005314(n).
%F A347493 G.f.: (1 - x)/(1 - x - x^2 - x^4).
%F A347493 Sum_{k=0..n} a(k)*F(n-k) = a(n+3) - F(n+2) for F(n)=A000045(n) the Fibonacci numbers.
%F A347493 5*a(n) = 2*(-1)^n + 3*A005314(n+1) -4*A005314(n) +2*A005314(n-1). - _R. J. Mathar_, Sep 30 2021
%t A347493 CoefficientList[Series[(1 - x)/(1 - x - x^2 - x^4), {x, 0, 40}], x] (* _Michael De Vlieger_, Mar 04 2022 *)
%t A347493 LinearRecurrence[{1,1,0,1},{1,0,1,1},60] (* _Harvey P. Dale_, Aug 17 2023 *)
%Y A347493 Cf. A005251, A005314, A060945.
%K A347493 nonn,easy
%O A347493 0,5
%A A347493 _Greg Dresden_ and _Yichen P. Wang_, Sep 03 2021