A294627 - OEIS

%I #68 Aug 31 2024 11:08:03

%S 0,1,2,3,7,14,27,55,110,219,439,878,1755,3511,7022,14043,28087,56174,

%T 112347,224695,449390,898779,1797559,3595118,7190235,14380471,

%U 28760942,57521883,115043767,230087534,460175067,920350135,1840700270,3681400539,7362801079,14725602158,29451204315

%N Expansion of x*(1 + x)/((1-2*x)*(1+x+x^2)).

%C A generalized tribonacci (A001590) sequence.

%C For n > 2, 6*a(n) is the number of quaternary sequences of length n in which all triples (q(i),q(i+1),q(i+2)) contain two (arbitrarily chosen) digits, e.g., 0 and 3.

%C Similarly, recurrences a(n) = a(n-1) + a(n-2) + k*a(n-3) are related to binary (k=0, the Fibonacci sequence A000045), ternary (k=1, the tribonacci sequence A001590), quinary (k=3) and so on sequences with all triples (t(i),t(i+1),t(i+2)) containing two (arbitrarily chosen) digits (usually 0 and k+1).

%C For n > 0, a(n) is the number of ways to tile a strip of length n with squares, dominoes, and two colors of trominoes, with the restriction that the first tile cannot be a tromino. - _Greg Dresden_ and Bora Bursalı, Aug 31 2023

%C For n > 1, a(n) is the number of ways to tile a strip of length n-2 with squares, dominoes, and two colors of trominoes, where the strip begins with an upper level of two cells. For example, when n=7 we have this strip of length 5:

%C  ___

%C |_|_|_____

%C |_|_|_|_|_|. - Guanji Chen and _Greg Dresden_, Jun 17 2024

%H Wojciech Florek, <a href="/A294627/b294627.txt">Table of n, a(n) for n = 0..500</a>

%H Wojciech Florek, <a href="http://doi.org/10.1016/j.amc.2018.06.014">A class of generalized Tribonacci sequences applied to counting problems</a>, Appl. Math. Comput., 338 (2018), 809-821.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,2).

%F a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n > 2.

%F a(n+1)/a(n) tends to 2, the unique real root of x^3 - x^2 - x - 2 = 0.

%F a(n+1) = abs(A078043(n)).

%F 7*a(n) = 3*2^n - A167373(n+1). - _R. J. Mathar_, Mar 24 2018

%F E.g.f.: exp(-x/2)*(9*exp(5*x/2) - 9*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/21. - _Stefano Spezia_, Aug 29 2024

%e For n=4 there are 6*7=42 quaternary sequences of length 4 such that each triple (i.e., exactly two of them: q1,q2,q3 and q2,q3,q4) contain both 0 and 3. They are 003x, 030x, 03y0, 0330, 330x, 303x, 30y3, 3003, 0y30, 3y03, y03x, y30x, where x=0,1,2,3 and y=1,2.

%t LinearRecurrence[{1, 1, 2}, {0, 1, 2}, 50] (* _Paolo Xausa_, Aug 28 2024 *)

%o (PARI) my(x='x+O('x^99)); concat(0, Vec(x*(1+x)/(1-x-x^2-2*x^3))) \\ _Altug Alkan_, Mar 03 2018

%Y Cf. A000045, A001590, A007040, A033129 (partial sums), A078043, A167373.

%K nonn,easy

%O 0,3

%A _Wojciech Florek_, Feb 12 2018