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.

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

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