A245963 Triangle read by rows: T(n,k) is the number of maximal hypercubes Q(p) in the Fibonacci cube Gamma(n) (i.e., Q(p) is an induced subgraph of Gamma(n) that is not a subgraph of a subgraph of Gamma(n) that is isomorphic to the hypercube Q(p+1)).

1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 3, 1, 0, 0, 1, 4, 0, 0, 0, 6, 1, 0, 0, 0, 4, 5, 0, 0, 0, 1, 10, 1, 0, 0, 0, 0, 10, 6, 0, 0, 0, 0, 5, 15, 1, 0, 0, 0, 0, 1, 20, 7, 0, 0, 0, 0, 0, 15, 21, 1, 0, 0, 0, 0, 0, 6, 35, 8, 0, 0, 0, 0, 0, 1, 35, 28, 1, 0, 0, 0, 0, 0, 0, 21, 56, 9, 0, 0, 0, 0, 0, 0, 7, 70, 36, 1

Offset

0,5

Comments

The nonzero entries in columns 0,1,2,... are rows 0,2,3,... of the Pascal triangle.

Row n contains 1+ceiling(n/2) entries.

Sum of entries in row n = A000931(n+6) (the Padovan sequence).

Sum_{k>=0}k*T(n,k) = A228364(n+1).

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10300 (Rows 1 <= n <= 200).

S. Klavzar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.

M. Mollard, Maximal hypercubes in Fibonacci and Lucas cubes, arXiv:1201.1494 [math.CO], 2012.

M. Mollard, Maximal hypercubes in Fibonacci and Lucas cubes, Discrete Appl. Math., 160, 2012, 2479-2483.

Formula

T(n,k) = binomial(k+1,n-2*k+1).

G.f.: (1+t*z*(1+z))/(1-t*(1+z)*z^2).

Example

Row 3 is 0,1,1. Indeed, the Fibonacci cube Gamma(3) is a square with an additional pendant edge attached to one of its vertices; the pendant edge is a maximal Q(1) and the square is a maximal Q(2).
Triangle starts:
  1;
  0, 1;
  0, 2;
  0, 1, 1;
  0, 0, 3;
  0, 0, 3, 1;
  0, 0, 1, 4;
  0, 0, 0, 6, 1;

Maple

T := proc (n, k) options operator, arrow: binomial(1+k, n-2*k+1) end proc: for n from 0 to 20 do seq(T(n, k), k = 0 .. (n+1)*(1/2)) end do; # yields sequence in triangular form

Mathematica

Table[Binomial[k + 1, n - 2 k + 1], {n, 0, 17}, {k, 0, Ceiling[n/2]}] // Flatten (* Michael De Vlieger, Jul 16 2017 *)

Crossrefs

Cf. A000931, A228364, A245964.

Sequence in context: A219023 A025427 A348536 * A291375 A346914 A033778

Adjacent sequences: A245960 A245961 A245962 * A245964 A245965 A245966

Keyword

nonn,tabf

Author

Emeric Deutsch, Aug 13 2014

Status

approved