OFFSET
0,1
COMMENTS
For n >= 5, the number a(n-3) is the dimension of a commutative Hecke algebra of affine type D_n with independent parameters. See Theorem 1.4, Corollary 1.5, and the table on page 524 in the link "Hecke algebras with independent parameters". - Jia Huang, Jan 20 2019
From Greg Dresden and Yiming Wu, Sep 10 2023: (Start)
For n >= 3, a(n) is the number of ways to tile this shape of length n+2 with squares and dominos:
_ _
_|_|___________________|_|_
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
|_| |_|. (End)
For n >= 3, a(n) is the number of edge covers of the kayak paddle graphs KP(3,3,n-3), where we interpret KP(3,3,0) as two C_3's with one common vertex. - Feryal Alayont, Sep 28 2024
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jia Huang, Hecke algebras with independent parameters, arXiv preprint arXiv:1405.1636 [math.RT], 2014; Journal of Algebraic Combinatorics 43 (2016) 521-551.
Eric Weisstein's World of Mathematics, Kayak Paddle Graph.
Index entries for linear recurrences with constant coefficients, signature (1,1).
FORMULA
a(n) = (5 + 2*sqrt(5)/5)*((1 + sqrt(5))/2)^n + (5 - 2*sqrt(5)/5)*((1 - sqrt(5))/2)^n. - Antonio Alberto Olivares, Jun 07 2011
a(n) = 7*Fibonacci(n) + 10*Fibonacci(n-1). - Charles R Greathouse IV, Jun 08 2011
G.f.: (10-3*x)/(1-x-x^2). - Colin Barker, Jan 11 2012
a(n) = 4*Fibonacci(n+1) + 3*LucasL(n). - G. C. Greubel, Oct 26 2022
MAPLE
seq(coeff(series((10-3*x)/(1-x-x^2), x, n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Jan 22 2019
MATHEMATICA
LinearRecurrence[{1, 1}, {10, 7}, 100]
PROG
(PARI) a(n)=7*fibonacci(n)+10*fibonacci(n-1) \\ Charles R Greathouse IV, Jun 08 2011
(Magma) [n le 2 select 13-3*n else Self(n-1)+Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012
(SageMath) [7*fibonacci(n+1) +3*fibonacci(n-1) for n in range(51)] # G. C. Greubel, Oct 26 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vladimir Joseph Stephan Orlovsky, Jun 07 2011
STATUS
approved