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 A128100 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)). 2
 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; text; internal format)
 OFFSET 0,3 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=0..floor(n/2)} k*T(n,k) = 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. - Philippe Deléham, Jan 24 2012 Riordan array (1/(1-x-x^2), x^2/(1-x-x^2)), with zeros omitted. - Philippe Deléham, Feb 06 2012 Diagonal sums are A000073(n+2) (tribonacci numbers). - Philippe Deléham, Feb 16 2014 Number of induced subgraphs of the Fibonacci cube Gamma(n-1) that are isomorphic to the hypercube Q_k. Example: row n=4 is 5, 5, 1; indeed, the Fibonacci cube Gamma(3) is a square with an additional pendant edge attached to one of its vertices; it has 5 vertices (i.e., Q_0's), 5 edges (i.e., Q_1's) and 1 square (i.e., Q_2). - Emeric Deutsch, Aug 12 2014 Row n gives the coefficients of the polynomial p(n,x) defined as the numerator of the rational function given by f(n,x) = 1 + (x + 1)/f(n-1,x), where f(x,0) = 1. Conjecture: for n > 2, p(n,x) is irreducible if and only if n is a (prime - 2). - Clark Kimberling, Oct 22 2014 LINKS C.-P. Chou and H. A. Witek, An Algorithm and FORTRAN Program for Automatic Computation of the Zhang-Zhang Polynomial of Benzenoids, MATCH: Commun. Math. Comput. Chem, 68 (2012) 3-30. See Eq. (9). - From N. J. A. Sloane, Dec 23 2012 S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, preprint. S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, Acta Appl. Math. 117, 2012, 93-105. - Emeric Deutsch, Aug 12 2014 FORMULA G.f.: 1/(1-z-(1+t)z^2). Sum_{k=0..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. - Philippe Deléham, Jan 24 2012 T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1). - Philippe Deléham, Jan 24 2012 G.f.: T(0)/2, where T(k) = 1 + 1/(1 - (2*k+1+ x*(1+y))*x/((2*k+2+ x*(1+y))*x + 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013 T(n,k) = Sum_{i=k..floor(n/2)} binomial(n-i,i)*binomial(i,k). See Corollary 3.3 in the Klavzar et al. link. - Emeric Deutsch, Aug 12 2014 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 - Philippe Deléham, Jan 24 2012 Here are the first 4 polynomials p(n,x) as in Comment and generated by Mathematica program: 1, 2 + x, 3 + 2x, 5 + 5 x + x^2. - Clark Kimberling, Oct 22 2014 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 MATHEMATICA p[x_, n_] := 1 + (x + 1)/p[x, n - 1]; p[x_, 1] = 1; Numerator[Table[Factor[p[x, n]], {n, 1, 20}]]  (* Clark Kimberling, Oct 22 2014 *) CROSSREFS Cf. A001045, A000045, A001629, A001628, A001872, A001873, A073371. Cf. A109466, A119473. Sequence in context: A034393 A068932 A151533 * A035579 A045931 A325193 Adjacent sequences:  A128097 A128098 A128099 * A128101 A128102 A128103 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Feb 18 2007 STATUS approved

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Last modified December 10 20:38 EST 2019. Contains 329909 sequences. (Running on oeis4.)