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 A278136 Triangle read by rows: T(n,k) is the maximum number of disjoint subgraphs of the Fibonacci cube Gamma(n) that are isomorphic to the hypercube of dimension k. 2
 1, 2, 1, 3, 1, 5, 2, 1, 8, 4, 1, 13, 6, 2, 1, 21, 10, 5, 1, 34, 17, 7, 2, 1, 55, 27, 12, 6, 1, 89, 44, 22, 8, 2, 1, 144, 72, 34, 14, 7, 1, 233, 116, 56, 28, 9, 2, 1, 377, 188, 94, 42, 16, 8, 1, 610, 305, 150, 70, 35, 10, 2, 1, 987, 493, 244, 122, 51, 18, 9, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of entries in row n is 1 + ceiling(n/2). T(n,0) = F(n+2) = A000045(n+2) (Fibonacci); number of vertices of Gamma(n). Sum of entries in row n is A278137(n). T(n,1) = floor(F(n+2)/2) (see Lemma 2.1 in the Gravier et al. paper). The generating function of column k is x^{2k-1}/((1-x^3)^k*(1-x-x^2)) (k>=0) (see Corollary 2.5 in the Gravier et al. paper). LINKS Indranil Ghosh, Rows 0..100, flattened S. Gravier, M. Mollard, S. Spacapan, S. S. Zemljic, On disjoint hypercubes in Fibonacci cubes, Discrete Appl. Math., 190-191, 2015, 50-55. S. Klavzar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522. M. Mollard, Maximal hypercubes in Fibonacci and Lucas cubes, Discrete Appl. Math., 160, 2012, 2479-2483. FORMULA T(n,k) = Sum_{i=k-1..floor((n+k-2)/3)} binomial(i,k-1)*F(n+k-3i-1), where F(j) = A000045(j) (Fibonacci); (see Corollary 2.4 in the Gravier et al. paper). T(n,k) = T(n-2,k-1) + T(n-3,k) (n>=3, k>=1) (see Theorem 2.2 in the Gravier et al. paper). EXAMPLE Row 3 is 5,2,1. Indeed, the Fibonacci cube Gamma(3) has 5 vertices A, B, C, D, E and edges AB, BC, CD, DA, DE and so it has at most 2 disjoint edges and it has one square. Triangle starts:    1;    2, 1;    3, 1;    5, 2, 1;    8, 4, 1;   13, 6, 2, 1; MAPLE with(combinat): F := proc (k) options operator, arrow: fibonacci(k) end proc; T := proc (n, k) options operator, arrow: sum(binomial(i, k-1)*F(n+k-3*i-1), i = k-1 .. floor((1/3)*n+(1/3)*k-2/3)) end proc: for n from 0 to 20 do seq(T(n, k), k = 0 .. ceil((1/2)*n)) end do; # yields sequence in triangular form MATHEMATICA Flatten[Table[Sum[Binomial[i, k-1] Fibonacci[n+k-3i-1], {i, k-1, Floor[(n+k-2)/3]}], {n, 0, 14}, {k, 0, Ceiling[n/2]}]] (* Indranil Ghosh, Mar 05 2017 *) CROSSREFS Cf. A000045, A278137. Sequence in context: A168018 A173238 A173284 * A085053 A296118 A296121 Adjacent sequences:  A278133 A278134 A278135 * A278137 A278138 A278139 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Feb 26 2017 STATUS approved

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Last modified August 1 21:02 EDT 2021. Contains 346408 sequences. (Running on oeis4.)