A100477 a(n) = 3*a(n-1) + 2*a(n-2) + a(n-3) if n>=3, otherwise a(n) = n.

0, 1, 2, 8, 29, 105, 381, 1382, 5013, 18184, 65960, 239261, 867887, 3148143, 11419464, 41422565, 150254766, 545028892, 1977018773, 7171368869, 26013173045, 94359275646, 342275541897, 1241558350028, 4503585409524

Offset: 0

Author

gamo (gamo(AT)telecable.es), Nov 22 2004

Keywords

• nonn
• easy

Comments

Weighted sum of the three previous terms.

a(n+1) is the number of ways to tile a strip of length n with 3 colors of squares, 2 colors of dominos, and 1 color of tromino, with the restriction that if the first tile is a square, then it can only use two colors. - Greg Dresden and Bora Bursali, Aug 17 2023

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,2,1).

Crossrefs

Cf. A108153.

Programs

Magma

[n le 3 select n-1 else 3*Self(n-1)+2*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, May 20 2015

Mathematica

RecurrenceTable[{a[n]== 3a[n-1] +2a[n-2] +a[n-3], a[0]==0, a[1]==1, a[2]==2}, a, {n,0,26}] (* or *)
CoefficientList[ Series[(x^2-x)/(x^3+2x^2+3x-1), {x,0,26}], x] (* Robert G. Wilson v, May 19 2015 *)
LinearRecurrence[{3,2,1},{0,1,2},40] (* Harvey P. Dale, Jun 19 2015 *)

Perl

#!/usr/local/bin/perl -w $d=0; $c=1; $b=2; print "$d,$c,$b,"; $a=0; for (;;){ $a=3*$b+2*$c+$d; $d=$c; $c=$b; $b=$a; print "$a,"; last if ($a >2**61); } _END_

SageMath

@CachedFunction
def a(n): # a = A100477
    if (n<3): return n
    else: return 3*a(n-1)+2*a(n-2)+a(n-3)
[a(n) for n in range(41)] # G. C. Greubel, Apr 06 2023

Formula

From R. J. Mathar, Aug 22 2008: (Start)

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

a(n) = A108153(n) - A108153(n-1). (End)

a(0)=0, a(1)=1, a(2)=2, a(n)=3*a(n-1)+2*a(n-2)+a(n-3). - Harvey P. Dale, Jun 19 2015