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

Author

N. J. A. Sloane, Dec 08 2004

Keywords

• sign
• easy

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).

Crossrefs

Cf. A232542, A232543.

Cf. A001590, A020992.

Cf. A000073.

Programs

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 *)

PARI

Vec(-(1-3*x-x^2)/(1-x-x^2-x^3) + O(x^70)) \\ Michel Marcus, Sep 25 2015

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

Extensions

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