OFFSET
0,2
COMMENTS
A tribonacci sequence.
From Greg Dresden and Veda Garigipati, Jun 14 2022: (Start)
For n >= 2, a(n+2) is the number of ways to tile this figure of length n with squares, dominoes, and "trominoes" (of length 3):
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|_|_|___________
|_|_|_|_|_|_|_|_|
As an example, here is one of the 254 possible tilings of this figure of length 8 with squares, dominoes, and trominoes:
___
| |_|___________
|_|_____|_|___|_|. (End)
LINKS
Robert Price, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
S. Kak, The Golden Mean and the Physics of Aesthetics, arXiv:physics/0411195 [physics.hist-ph], 2004.
Eric Weisstein's World of Mathematics, Tribonacci Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1).
FORMULA
O.g.f.: -(1-3x-x^2)/(1-x-x^2-x^3). - R. J. Mathar, Aug 22 2008
a(n) = T(n-2) + T(n) + T(n+1) where T(n) = A000073(n) the tribonacci sequence, for n >= 2. - Greg Dresden and Veda Garigipati, Jun 14 2022
MAPLE
a[0]:=-1:a[1]:=2:a[2]:=2:for n from 3 to 42 do a[n]:=a[n-1]+a[n-2]+a[n-3] od: seq(a[n], n=0..42);
MATHEMATICA
a[0] = -1; a[1] = a[2] = 2; a[n_] := a[n] = a[n - 1] + a[n - 2] + a[n - 3]; Table[ a[n], {n, 0, 35}] (* Robert G. Wilson v, Dec 09 2004 *)
LinearRecurrence[{1, 1, 1}, {-1, 2, 2}, 34] (* Ray Chandler, Dec 08 2013 *)
PROG
(PARI) Vec(-(1-3*x-x^2)/(1-x-x^2-x^3) + O(x^70)) \\ Michel Marcus, Sep 25 2015
CROSSREFS
KEYWORD
sign,easy
AUTHOR
N. J. A. Sloane, Dec 08 2004
EXTENSIONS
STATUS
approved