OFFSET
0,3
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
E. Czabarka et al, Enumerations of peaks and valleys on non-decreasing Dyck paths, Disc. Math. 341 (2018) 2789-2807. See Table 4.
Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 22.
Index entries for linear recurrences with constant coefficients, signature (5,-7,2).
FORMULA
G.f.: x(1-x)/((1-2x)(1-3x+x^2)).
a(n) = sum{k=0..n+1, binomial(n+1, k+1)*sum{j=0..floor(k/2), F(k-2j)}}.
a(n) = 5*a(n-1)-7*a(n-2)+2*a(n-3) for n>2. - Colin Barker, Sep 12 2016
MATHEMATICA
Table[Fibonacci[2n+2]-2^n, {n, 0, 30}] (* or *) LinearRecurrence[{5, -7, 2}, {0, 1, 4}, 40] (* Harvey P. Dale, Jul 21 2016 *)
PROG
(Magma) [Fibonacci(2*n+2)-2^n: n in [0..30]]; // Vincenzo Librandi, Apr 21 2011
(PARI) concat(0, Vec(x*(1-x)/((1-2*x)*(1-3*x+x^2)) + O(x^40))) \\ Colin Barker, Sep 12 2016
(PARI) a(n)=fibonacci(2*n+2)-2^n \\ Charles R Greathouse IV, Sep 12 2016
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 17 2005
STATUS
approved