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A278137 Maximum number of disjoint subgraphs of the Fibonacci cube Gamma(n) that are isomorphic to the hypercube of dimension k, summed over all k. 2
1, 3, 4, 8, 13, 22, 37, 61, 101, 166, 272, 445, 726, 1183, 1925, 3129, 5082, 8248, 13379, 21692, 35157, 56963, 92271, 149434, 241970, 391755, 634190, 1026561, 1661567, 2689209, 4352208, 7043314, 11398035, 18444678, 29847123, 48297643, 78152505, 126460400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

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.

Index entries for linear recurrences with constant coefficients, signature (1,2,0,-2,-1).

FORMULA

a(n) = a(n-2) + a(n-3) + 2*F(n), where F(n) = A000045(n) (Fibonacci); a(0)=1, a(1)=3, a(2)=4; follows from Theorem 2.2 of the Gravier et al. paper.

a(n) = Sum(A278136(n,k), k>=0).

From Colin Barker, Feb 26 2017: (Start)

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-4) - a(n-5) for n>4.

G.f.: (1 + 2*x - x^2 - 2*x^3 - x^4) / ((1 - x - x^2)*(1 - x^2 - x^3)).

(End)

EXAMPLE

a(3) = 8; indeed, row 3 of A278136 is 5,2,1.

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: seq(add(T(n, k), k = 0 .. ceil((1/2)*n)), n = 0 .. 45);

with(combinat): a := proc (n) if n = 0 then 1 elif n = 1 then 3 elif n = 2 then 4 else a(n-2)+a(n-3)+2*fibonacci(n) end if end proc: seq(a(n), n = 0 .. 45);

PROG

(PARI) Vec((1 + 2*x - x^2 - 2*x^3 - x^4) / ((1 - x - x^2)*(1 - x^2 - x^3)) + O(x^40)) \\ Colin Barker, Feb 27 2017

CROSSREFS

Cf. A000045, A278136.

Sequence in context: A049720 A078172 A022308 * A206268 A178749 A121980

Adjacent sequences:  A278134 A278135 A278136 * A278138 A278139 A278140

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Feb 26 2017

STATUS

approved

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Last modified October 23 03:21 EDT 2019. Contains 328335 sequences. (Running on oeis4.)