OFFSET
0,3
COMMENTS
With offset 4, number of 132-avoiding two-stack sortable permutations which contain exactly one subsequence of type 51234.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Hung Viet Chu, Partial Sums of the Fibonacci Sequence, arXiv:2106.03659 [math.CO], 2021.
Ligia Loretta Cristea, Ivica Martinjak, and Igor Urbiha, Hyperfibonacci Sequences and Polytopic Numbers, arXiv:1606.06228 [math.CO], 2016.
E. S. Egge and T. Mansour, 132-avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and Pell Numbers, arXiv:math/0205206 [math.CO], 2002.
T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).
FORMULA
a(n) = Sum_{k=0..n} A000045(n-k)*k*(k+1)/2. - Benoit Cloitre, Jan 06 2003
G.f.: x/((1-x)^3*(1-x-x^2)).
From Paul Barry, Oct 07 2004: (Start)
a(n-2) = Sum_{k=0..floor(n/2)} binomial(n-k, k+3).
a(n-2) = Sum_{k=0..n} binomial(k, n-k+3). (End)
a(n) = Fibonacci(n+6) - (n^2 + 7*n + 16)/2.
MAPLE
with(combinat); seq(fibonacci(n+6)-(n^2+7*n+16)*(1/2), n = 0..40); # G. C. Greubel, Sep 05 2019
MATHEMATICA
Nest[Accumulate, Fibonacci[Range[0, 30]], 3] (* or *) LinearRecurrence[{4, -5, 1, 2, -1}, {0, 1, 4, 11, 25}, 40] (* Harvey P. Dale, Aug 19 2017 *)
PROG
(PARI) a(n)=fibonacci(n+6)-n*(n+7)/2-8 \\ Charles R Greathouse IV, Jun 11 2015
(Magma) [Fibonacci(n+6) - (n^2 + 7*n + 16)/2: n in [0..40]]; // G. C. Greubel, Sep 05 2019
(Sage) [fibonacci(n+6) - (n^2 + 7*n + 16)/2 for n in (0..40)] # G. C. Greubel, Sep 05 2019
(GAP) List([0..40], n-> Fibonacci(n+6) - (n^2 + 7*n + 16)/2); # G. C. Greubel, Sep 05 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved