A014166 Apply partial sum operator 4 times to Fibonacci numbers.
0, 1, 5, 16, 41, 92, 189, 365, 674, 1204, 2098, 3588, 6050, 10093, 16703, 27476, 44995, 73440, 119575, 194345, 315460, 511576, 829060, 1342936, 2174596, 3520457, 5698329, 9222440, 14924829, 24151764, 39081553
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.
- Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).
Programs
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GAP
List([0..30], n-> Fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6); # G. C. Greubel, Sep 06 2019
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Magma
[Fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6: n in [0..30]]; // G. C. Greubel, Sep 06 2019
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Maple
with(combinat); seq(fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6, n = 0..30); # G. C. Greubel, Sep 06 2019
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Mathematica
Nest[Accumulate, Fibonacci[Range[0, 30]], 4] (* Jean-François Alcover, Jan 08 2019 *)
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PARI
a(n)=fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6 \\ Charles R Greathouse IV, Jun 11 2015
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Sage
[fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6 for n in (0..30)] # G. C. Greubel, Sep 06 2019
Formula
a(n) = Fibonacci(n+8) - (n^3 +12*n^2 +59*n +126)/6.
G.f.: x/((1-x)^4*(1-x-x^2)).