A014162 Apply partial sum operator thrice to Fibonacci numbers.

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

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)

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

Cf. A000045, A001924, A228074.

Right-hand column 6 of triangle A011794.

Sequence in context: A193912 A136395 A014160 * A014169 A113684 A356619

Adjacent sequences: A014159 A014160 A014161 * A014163 A014164 A014165

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved