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
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).
Programs
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GAP
List([0..40], n-> Fibonacci(n+6) - (n^2 + 7*n + 16)/2); # G. C. Greubel, Sep 05 2019
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Magma
[Fibonacci(n+6) - (n^2 + 7*n + 16)/2: n in [0..40]]; // G. C. Greubel, Sep 05 2019
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Maple
with(combinat); seq(fibonacci(n+6)-(n^2+7*n+16)*(1/2), n = 0..40); # G. C. Greubel, Sep 05 2019
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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 *) -
PARI
a(n)=fibonacci(n+6)-n*(n+7)/2-8 \\ Charles R Greathouse IV, Jun 11 2015
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Sage
[fibonacci(n+6) - (n^2 + 7*n + 16)/2 for n in (0..40)] # G. C. Greubel, Sep 05 2019
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.
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