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
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Feb 26 2017
STATUS
approved