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.

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

njas

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