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A002478
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Bisection of A000930.
(Formerly M2572 N1017)
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16
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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; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 18 2010: (Start)
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. (End)
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REFERENCES
| 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).
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LINKS
| 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.
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FORMULA
| 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
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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.
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MAPLE
| A002478:=-1/(-1+z+2*z**2+z**3); [S. Plouffe in his 1992 dissertation.]
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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 ]
LinearRecurrence[{1, 2, 1}, {1, 1, 3}, 40] (* From Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
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CROSSREFS
| Cf. A054856, A054857, A025234.
Cf. A078007.
Cf. A078039.
Sequence in context: A103788 A106461 A095768 * A106496 A052933 A071014
Adjacent sequences: A002475 A002476 A002477 * A002479 A002480 A002481
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KEYWORD
| easy,nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments from Silvia Heubach (silvi(AT)cine.net), Apr 21 2000
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