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A128100
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Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0<=k<=floor(n/2)).
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0
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1, 1, 2, 1, 3, 2, 5, 5, 1, 8, 10, 3, 13, 20, 9, 1, 21, 38, 22, 4, 34, 71, 51, 14, 1, 55, 130, 111, 40, 5, 89, 235, 233, 105, 20, 1, 144, 420, 474, 256, 65, 6, 233, 744, 942, 594, 190, 27, 1, 377, 1308, 1836, 1324, 511, 98, 7, 610, 2285, 3522, 2860, 1295, 315, 35, 1, 987, 3970
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row sums are the Jacobsthal numbers (A001045). Column 0 yields the Fibonacci numbers (A000045); the other columns yield convolved Fibonacci numbers (A001629,A001628,A001872,A001873, etc.). Sum(k(T(n,k),k=0..floor(n/2))=A073371(n-2).
Triangle T(n,k), with zeros omitted, given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - DELEHAM Philippe, Jan 24 2012
Riordan array (1/(1-x-x^2), x^2/(1-x-x^2)), with zeros omitted. - DELEHAM Philippe, Feb 06 2012
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FORMULA
| G.f.=1/[1-z-(1+t)z^2].
Sum_{k, 0<=k<=n} T(n,k)*x^k = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1), A000012(n), A010892(n), A107920(n+1), A106852(n), A106853(n), A106854(n), A145934(n), A145976(n), A145978(n), A146078(n), A146080(n), A146083(n), A146084(n) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, and -13, respectively. - DELEHAM Philippe, Jan 24 2012
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1). - DELEHAM Philippe, Jan 24 2012
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EXAMPLE
| Triangle starts:
1;
1;
2,1;
3,2;
5,5,1;
8,10,3;
13,20,9,1;
21,38,22,4;
Triangle (1, 1, -1, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, ...) begins :
1
1, 0
2, 1, 0
3, 2, 0, 0
5, 5, 1, 0, 0
8, 10, 3, 0, 0, 0
13, 20, 9, 1, 0, 0, 0
21, 38, 22, 4, 0, 0, 0, 0 - DELEHAM Philippe, Jan 24 2012
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MAPLE
| G:=1/(1-z-(1+t)*z^2): Gser:=simplify(series(G, z=0, 19)): for n from 0 to 16 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 16 do seq(coeff(P[n], t, j), j=0..floor(n/2)) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A001045, A000045, A001629, A001628, A001872, A001873, A073371.
Cf. A109466, A119473.
Sequence in context: A034393 A068932 A151533 * A035579 A045931 A079974
Adjacent sequences: A128097 A128098 A128099 * A128101 A128102 A128103
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KEYWORD
| nonn,tabf,changed
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 18 2007
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