This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A014166 #49 Feb 03 2025 13:51:36 %S A014166 0,1,5,16,41,92,189,365,674,1204,2098,3588,6050,10093,16703,27476, %T A014166 44995,73440,119575,194345,315460,511576,829060,1342936,2174596, %U A014166 3520457,5698329,9222440,14924829,24151764,39081553 %N A014166 Apply partial sum operator 4 times to Fibonacci numbers. %H A014166 G. C. Greubel, <a href="/A014166/b014166.txt">Table of n, a(n) for n = 0..1000</a> %H A014166 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 A014166 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 A014166 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,6,1,-3,1). %F A014166 a(n) = Fibonacci(n+8) - (n^3 +12*n^2 +59*n +126)/6. %F A014166 G.f.: x/((1-x)^4*(1-x-x^2)). %p A014166 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 A014166 Nest[Accumulate, Fibonacci[Range[0, 30]], 4] (* _Jean-François Alcover_, Jan 08 2019 *) %o A014166 (PARI) a(n)=fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6 \\ _Charles R Greathouse IV_, Jun 11 2015 %o A014166 (Magma) [Fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6: n in [0..30]]; // _G. C. Greubel_, Sep 06 2019 %o A014166 (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 A014166 (GAP) List([0..30], n-> Fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6); # _G. C. Greubel_, Sep 06 2019 %Y A014166 Cf. A000045, A228074, A136431. %Y A014166 Right-hand column 8 of triangle A011794. %K A014166 nonn,easy %O A014166 0,3 %A A014166 _N. J. A. Sloane_