A020992 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 0, a(1) = 2, a(2) = 1.

0, 2, 1, 3, 6, 10, 19, 35, 64, 118, 217, 399, 734, 1350, 2483, 4567, 8400, 15450, 28417, 52267, 96134, 176818, 325219, 598171, 1100208, 2023598, 3721977, 6845783, 12591358, 23159118, 42596259, 78346735, 144102112, 265045106, 487493953, 896641171, 1649180230

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Tribonacci sequence beginning 0, 2, 1.

Pisano period lengths: 1, 4, 13, 8, 31, 52, 48, 16, 39, 124, 110, 104, 168, 48, 403, 32, 96, 156, 360, 248,.... - R. J. Mathar, Aug 10 2012

One bisection is 0, 1, 6, 19, 64, 217, 734, 2483, 8400,.. and the other 2, 3, 10, 35, 118, 399, 1350, 4567,... both with recurrence b(n)=3*b(n-1)+b(n-2)+b(n-3). - R. J. Mathar, Aug 10 2012

From Greg Dresden and Jiarui Zhou, Jun 30 2025: (Start)

For n >= 4, 2*a(n) is the number of ways to tile this shape of length n-2 with squares, dominos, and trominos (of length 3):

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As an example, here is one of the 2*a(10) = 434 ways to tile this shape of length 8:

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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.
• Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Crossrefs

Cf. A000032, A000073, A001590, A232498, A233554.

Programs

Magma

I:=[0,2,1]; [n le 3 select I[n] else Self(n-1) + Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Feb 09 2018

Mathematica

LinearRecurrence[{1,1,1},{0,2,1},100] (* Vladimir Joseph Stephan Orlovsky, Jun 07 2011 *)

PARI

my(x='x+O('x^30)); concat([0], Vec(x*(2-x)/(1-x-x^2-x^3))) \\ G. C. Greubel, Feb 09 2018

Formula

G.f.: x*(2-x)/(1-x-x^2-x^3).

a(n) = 2*A000073(n+1)-A000073(n). - R. J. Mathar, Aug 22 2008

a(n) = 2*a(n-1) - a(n-4), n>3. - Vincenzo Librandi, Jun 08 2011