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

1, 0, 1, 1, 3, 4, 8, 13, 24, 41, 73, 127, 224, 392, 689, 1208, 2121, 3721, 6531, 11460, 20112, 35293, 61936, 108689, 190737, 334719, 587392, 1030800, 1808929, 3174448, 5570769, 9776017, 17155715, 30106180, 52832664, 92714861, 162703240, 285524281, 501060185, 879299327, 1543062752

Offset

0,5

Comments

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).

Links

Michael De Vlieger, Table of n, a(n) for n = 0..4096

Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck Paths with catastrophes modulo the positions of a given pattern, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.

Index entries for linear recurrences with constant coefficients, signature (1,1,0,1).

Formula

a(n) = 2*A060945(n) - A005251(n) - A005314(n).

G.f.: (1 - x)/(1 - x - x^2 - x^4).

Sum_{k=0..n} a(k)*F(n-k) = a(n+3) - F(n+2) for F(n)=A000045(n) the Fibonacci numbers.

5*a(n) = 2*(-1)^n + 3*A005314(n+1) -4*A005314(n) +2*A005314(n-1). - R. J. Mathar, Sep 30 2021

Mathematica

CoefficientList[Series[(1 - x)/(1 - x - x^2 - x^4), {x, 0, 40}], x] (* Michael De Vlieger, Mar 04 2022 *)
LinearRecurrence[{1, 1, 0, 1}, {1, 0, 1, 1}, 60] (* Harvey P. Dale, Aug 17 2023 *)

Crossrefs

Cf. A005251, A005314, A060945.

Sequence in context: A206268 A178749 A121980 * A091231 A250473 A049893

Adjacent sequences: A347490 A347491 A347492 * A347494 A347495 A347496

Keyword

nonn,easy

Author

Greg Dresden and Yichen P. Wang, Sep 03 2021

Status

approved