0,3

Number of ways to tile a 3 X n region with 1 X 1, 2 X 2 and 3 X 3 tiles.

Diagonal sums of A063967. - Paul Barry (pbarry(AT)wit.ie), Nov 09 2005

Row sums of number triangle A116088. - Paul Barry (pbarry(AT)wit.ie), Feb 04 2006

Sequence is identical to its second differences negated, minus the first 3 terms. - Paul Curtz (bpcrtz(AT)free.fr), Feb 10 2008

a(n) = term (3,3) in the 3x3 matrix [0,1,0; 0,0,1; 1,2,1]^n. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 30 2008

a(n)/a(n-1) tends to 2.147899035..., an eigenvalue of the matrix and a root to x^3 - x^2 - 2x - 1 = 0. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 30 2008

E. Deutsch, Counting tilings with L-tiles and squares, Problem 10877, Amer. Math. Monthly, 110 (March 2003), 245-246.

L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 322.

S. Heubach, Tiling an m X n Area with Squares of Size up to k X k (m<=5), Congressus Numerantium 140 (1999), pp. 43-64.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..300

L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 412

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

a(n)=a(n-1)+2a(n-2)+a(n-3)

a(n)=sum{k=0..n, binomial(2n-2k, k)} - Paul Barry (pbarry(AT)wit.ie), Nov 13 2004

a(n)=sum{k=0..floor(n/2), sum{j=0..n-k, C(j, n-k-j)C(j, k)}}; - Paul Barry (pbarry(AT)wit.ie), Nov 09 2005

a(n)=sum{k=0..n, C(2k,n-k)}=sum{k=0..n, C(n,k)C(3k,n)/C(3k,k)}. - Paul Barry (pbarry(AT)wit.ie), Feb 04 2006

a(3)=6 as there is one tiling of a 3 X 3 region with only 1 X 1 tiles, 4 tilings with exactly one 2 X 2 tile and 1 tiling consisting of the 3 X 3 tile.

A002478:=-1/(-1+z+2*z**2+z**3); [S. Plouffe in his 1992 dissertation.]

f[ A_ ] := Module[ {til = A}, AppendTo[ til, A[ [ -1 ] ] + 2A[ [ -2 ] ] + A[ [ -3 ] ] ] ]; NumOfTilings[ n_ ] := Nest[ f, {1, 1, 3}, n - 2 ]; NumOfTilings[ 30 ]

Cf. A054856, A054857, A025234.

Cf. A078007.

Cf. A078039.

Sequence in context: A103788 A106461 A095768 this_sequence A106496 A052933 A071014

Adjacent sequences: A002475 A002476 A002477 this_sequence A002479 A002480 A002481

easy,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Silvia Heubach (silvi(AT)cine.net), Apr 21 2000