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Tetranacci sequence beginning with 3, 7, 12, 24.
1

%I #34 Aug 30 2024 09:53:10

%S 3,7,12,24,46,89,171,330,636,1226,2363,4555,8780,16924,32622,62881,

%T 121207,233634,450344,868066,1673251,3225295,6216956,11983568,

%U 23099070,44524889,85824483,165432010,318880452,614661834,1184798779,2283773075,4402114140,8485347828

%N Tetranacci sequence beginning with 3, 7, 12, 24.

%C By "Tetranacci sequence" we mean a sequence in which each term is the sum of the four previous terms.

%C For n>1, a(n) is the number of ways to tile this figure of length n with squares, dominoes, trominoes, and tetraminoes:

%C ___

%C |_|_|_________ _

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

%H Paolo Xausa, <a href="/A356807/b356807.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1).

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

%F a(n) = 5*b(n+2) + 2*b(n+1) - 2*b(n-2) for b(n) = A000078(n) the tetranacci numbers.

%F a(n) = L(n+2) - F(n-2) + Sum_{k=0..n-3} a(k)*F(n-k-1), for L(n) and F(n) the Lucas and Fibonacci numbers.

%F G.f.: x*(-2*x^3 - 2*x^2 - 4*x - 3)/(x^4 + x^3 + x^2 + x - 1). - _Chai Wah Wu_, Aug 30 2022

%e Here is one of the a(6) = 89 ways to tile this figure of length 6 with tiles of length <= 4, this one using three squares, one domino, and one tromino:

%e ___

%e | |_|_______

%e |_|_____|_|_|

%t LinearRecurrence[{1, 1, 1, 1}, {3, 7, 12, 24}, 50] (* _Paolo Xausa_, Aug 30 2024 *)

%Y Cf. A022120, A100683.

%K nonn,easy

%O 1,1

%A _Greg Dresden_ and Hangyu Liang, Aug 29 2022