A100683 a(n) = a(n-1) + a(n-2) + a(n-3); a(0) = -1, a(1) = 2, a(2) = 2.

-1, 2, 2, 3, 7, 12, 22, 41, 75, 138, 254, 467, 859, 1580, 2906, 5345, 9831, 18082, 33258, 61171, 112511, 206940, 380622, 700073, 1287635, 2368330, 4356038, 8012003, 14736371, 27104412, 49852786, 91693569, 168650767, 310197122

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):

___

|_|_|___________

|_|_|_|_|_|_|_|_|

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

a(n+1) = 2*A001590(n+1) + A020992(n). - Creighton Dement, May 02 2005

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

Cf. A232542, A232543.

Cf. A001590, A020992.

Cf. A000073.

Sequence in context: A179283 A234850 A127165 * A153940 A049905 A167348

Adjacent sequences: A100680 A100681 A100682 * A100684 A100685 A100686

Keyword

sign,easy

Author

N. J. A. Sloane, Dec 08 2004

Extensions

More terms from Emeric Deutsch, Farideh Firoozbakht and Robert G. Wilson v, Dec 08 2004

Status

approved