A014166 - OEIS

%I #49 Feb 03 2025 13:51:36

%S 0,1,5,16,41,92,189,365,674,1204,2098,3588,6050,10093,16703,27476,

%T 44995,73440,119575,194345,315460,511576,829060,1342936,2174596,

%U 3520457,5698329,9222440,14924829,24151764,39081553

%N Apply partial sum operator 4 times to Fibonacci numbers.

%H G. C. Greubel, <a href="/A014166/b014166.txt">Table of n, a(n) for n = 0..1000</a>

%H Hung Viet Chu, <a href="https://arxiv.org/abs/2106.03659">Partial Sums of the Fibonacci Sequence</a>, arXiv:2106.03659 [math.CO], 2021.

%H Ligia Loretta Cristea, Ivica Martinjak, and Igor Urbiha, <a href="http://arxiv.org/abs/1606.06228">Hyperfibonacci Sequences and Polytopic Numbers</a>, arXiv:1606.06228 [math.CO], 2016.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,6,1,-3,1).

%F a(n) = Fibonacci(n+8) - (n^3 +12*n^2 +59*n +126)/6.

%F G.f.: x/((1-x)^4*(1-x-x^2)).

%p with(combinat); seq(fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6, n = 0..30); # _G. C. Greubel_, Sep 06 2019

%t Nest[Accumulate, Fibonacci[Range[0, 30]], 4] (* _Jean-François Alcover_, Jan 08 2019 *)

%o (PARI) a(n)=fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6 \\ _Charles R Greathouse IV_, Jun 11 2015

%o (Magma) [Fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6: n in [0..30]]; // _G. C. Greubel_, Sep 06 2019

%o (Sage) [fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6 for n in (0..30)] # _G. C. Greubel_, Sep 06 2019

%o (GAP) List([0..30], n-> Fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6); # _G. C. Greubel_, Sep 06 2019

%Y Cf. A000045, A228074, A136431.

%Y Right-hand column 8 of triangle A011794.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_