A033505 Expansion of 1/(1 - 3*x - x^2 + x^3).

1, 3, 10, 32, 103, 331, 1064, 3420, 10993, 35335, 113578, 365076, 1173471, 3771911, 12124128, 38970824, 125264689, 402640763, 1294216154, 4160024536, 13371648999, 42980755379, 138153890600, 444070778180, 1427385469761, 4588073296863, 14747534582170

Offset

0,2

Comments

From Greg Dresden, Jun 25 2022: (Start)

a(n) is the number of ways to tile, with squares and dominoes, a 2 X n board with one extra space at the end. Here is the board for n=3:

_____

|_|_|_|_

|_|_|_|_|

and here is one of the a(3)=32 possible tilings of this board:

_____

| |_|_|_

|_|_|___|

(End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 4.

N. J. A. Sloane Notes on A030186 and A033505

Index entries for linear recurrences with constant coefficients, signature (3,1,-1).

Formula

a(n) = 3*a(n-1) + a(n-2) - a(n-3). - Greg Dresden, Aug 16 2018

Maple

seq(coeff(series(1/(1-3*x-x^2+x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 14 2019

Mathematica

CoefficientList[Series[1/(1-3x-x^2+x^3), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, 1, -1}, {1, 3, 10}, 30] (* Vincenzo Librandi, Aug 17 2018 *)

Prog

(Magma) I:=[1, 3, 10]; [n le 3 select I[n] else 3*Self(n-1)+Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Aug 17 2018
(PARI) my(x='x+O('x^30)); Vec(1/(1-3*x-x^2+x^3)) \\ G. C. Greubel, Oct 14 2019
(Sage)
def A033505_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P(1/(1-3*x-x^2+x^3)).list()
A033505_list(30) # G. C. Greubel, Oct 14 2019
(GAP) a:=[1, 3, 10];; for n in [4..30] do a[n]:=3*a[n-1]+a[n-2]-a[n-3]; od; a; # G. C. Greubel, Oct 14 2019

Crossrefs

Partial sums of A030186.

Sequence in context: A077826 A292398 A273351 * A297067 A063782 A071718

Adjacent sequences: A033502 A033503 A033504 * A033506 A033507 A033508

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 13 2002

Status

approved