|
%I #49 Jul 01 2022 12:20:15
%S -1,2,2,3,7,12,22,41,75,138,254,467,859,1580,2906,5345,9831,18082,
%T 33258,61171,112511,206940,380622,700073,1287635,2368330,4356038,
%U 8012003,14736371,27104412,49852786,91693569,168650767,310197122
%N a(n) = a(n-1) + a(n-2) + a(n-3); a(0) = -1, a(1) = 2, a(2) = 2.
%C A tribonacci sequence.
%C From _Greg Dresden_ and _Veda Garigipati_, Jun 14 2022: (Start)
%C 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):
%C ___
%C |_|_|___________
%C |_|_|_|_|_|_|_|_|
%C As an example, here is one of the 254 possible tilings of this figure of length 8 with squares, dominoes, and trominoes:
%C ___
%C | |_|___________
%C |_|_____|_|___|_|. (End)
%H Robert Price, <a href="/A100683/b100683.txt">Table of n, a(n) for n = 0..1000</a>
%H Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H S. Kak, <a href="http://arXiv.org/abs/physics/0411195">The Golden Mean and the Physics of Aesthetics</a>, arXiv:physics/0411195 [physics.hist-ph], 2004.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TribonacciNumber.html">Tribonacci Number</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1).
%F a(n+1) = 2*A001590(n+1) + A020992(n). - _Creighton Dement_, May 02 2005
%F O.g.f.: -(1-3x-x^2)/(1-x-x^2-x^3). - _R. J. Mathar_, Aug 22 2008
%F 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
%p 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);
%t 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 *)
%t LinearRecurrence[{1,1,1},{-1,2,2},34] (* _Ray Chandler_, Dec 08 2013 *)
%o (PARI) Vec(-(1-3*x-x^2)/(1-x-x^2-x^3) + O(x^70)) \\ _Michel Marcus_, Sep 25 2015
%Y Cf. A232542, A232543.
%Y Cf. A001590, A020992.
%Y Cf. A000073.
%K sign,easy
%O 0,2
%A _N. J. A. Sloane_, Dec 08 2004
%E More terms from _Emeric Deutsch_, _Farideh Firoozbakht_ and _Robert G. Wilson v_, Dec 08 2004
|
