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A014162 Apply partial sum operator thrice to Fibonacci numbers. 13
0, 1, 4, 11, 25, 51, 97, 176, 309, 530, 894, 1490, 2462, 4043, 6610, 10773, 17519, 28445, 46135, 74770, 121115, 196116, 317484, 513876, 831660, 1345861, 2177872, 3524111, 5702389, 9226935, 14929789 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
With offset 4, number of 132-avoiding two-stack sortable permutations which contain exactly one subsequence of type 51234.
LINKS
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.
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)
Convolution of A000045 and A000217 (Fibonacci and triangular numbers). - Ross La Haye, Nov 08 2004
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
Right-hand column 6 of triangle A011794.
Sequence in context: A193912 A136395 A014160 * A014169 A113684 A356619
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified May 18 15:59 EDT 2024. Contains 372664 sequences. (Running on oeis4.)