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.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 15, 17, 19.
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