|
|
A002478
|
|
Bisection of A000930.
(Formerly M2572 N1017)
|
|
54
|
|
|
1, 1, 3, 6, 13, 28, 60, 129, 277, 595, 1278, 2745, 5896, 12664, 27201, 58425, 125491, 269542, 578949, 1243524, 2670964, 5736961, 12322413, 26467299, 56849086, 122106097, 262271568, 563332848, 1209982081, 2598919345, 5582216355, 11990037126, 25753389181
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Number of ways to tile a 3 X n region with 1 X 1, 2 X 2 and 3 X 3 tiles.
Number of ternary words with subwords (0,0), (0,1) and (1,1) not allowed. - Olivier Gérard, Aug 28 2012
Sequence is identical to its second differences negated, minus the first 3 terms. - Paul Curtz, Feb 10 2008
a(n) = term (3,3) in the 3 X 3 matrix [0,1,0; 0,0,1; 1,2,1]^n. - Gary W. Adamson, 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, May 30 2008
INVERT transform of (1, 2, 1, 0, 0, 0, ...) = (1, 3, 6, 13, 28, ...); such that (1, 2, 1, 0, 0, 0, ...) convolved with (1, 1, 3, 6, 13, 28, 0, 0, 0, ...) shifts to the left. - Gary W. Adamson, Apr 18 2010
a(n) is the top left entry in the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 0, 1; 1, 0, 0] or of the 3 X 3 matrix [1, 1, 1; 1, 0, 0; 1, 1, 0]. - R. J. Mathar, Feb 03 2014
|
|
REFERENCES
|
Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.
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).
|
|
LINKS
|
|
|
FORMULA
|
G.f.: 1 / (1-x-2*x^2-x^3). [Simon Plouffe in his 1992 dissertation.]
a(n) = a(n-1) + 2*a(n-2) + a(n-3).
a(n) = Sum_{k=0..n} binomial(2*n-2*k, k). - Paul Barry, 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, Nov 09 2005
a(n) = Sum_{k=0..n} C(2*k,n-k) = Sum_{k=0..n} C(n,k)*C(3*k,n)/C(3*k,k). - Paul Barry, Feb 04 2006
|
|
EXAMPLE
|
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.
|
|
MATHEMATICA
|
f[A_]:= Module[{til = A}, AppendTo[til, A[[-1]] + 2A[[-2]] + A[[-3]]]]; NumOfTilings[ n_ ]:= Nest[ f, {1, 1, 3}, n-2]; NumOfTilings[30]
|
|
PROG
|
(Magma) I:=[1, 1, 3]; [n le 3 select I[n] else Self(n-1) +2*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Apr 14 2023
(SageMath)
@CachedFunction
if (n<3): return (1, 1, 3)[n]
else: return sum(binomial(2, j)*a(n-j) for j in range(1, 4))
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn,nice
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Additional comments from Silvia Heubach (silvi(AT)cine.net), Apr 21 2000
|
|
STATUS
|
approved
|
|
|
|