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 A078027 #44 Jan 10 2025 04:39:14
%S A078027 1,-1,1,0,0,1,0,1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,
%T A078027 200,265,351,465,616,816,1081,1432,1897,2513,3329,4410,5842,7739,
%U A078027 10252,13581,17991,23833,31572,41824,55405,73396,97229,128801,170625,226030,299426
%N A078027 Expansion of (1 - x)/(1 - x^2 - x^3).
%H A078027 G. C. Greubel, <a href="/A078027/b078027.txt">Table of n, a(n) for n = 0..1000</a>
%H A078027 P. Chinn and S. Heubach, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Heubach/heubach5.html">Integer Sequences Related to Compositions without 2's</a>, J. Integer Seqs., 6 (2003), #03.2.3.
%H A078027 Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, <a href="https://doi.org/10.9734/ajpas/2024/v26i12691">A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers</a>, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
%H A078027 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1).
%F A078027 a(n) is asymptotic to r^(n-2) / (2*r+3) where r = 1.3247179572447..., the real root of x^3 = x + 1. For n >= 4, a(n) = a(n-2) + a(n-3). - _Philippe Deléham_, Jan 13 2004
%F A078027 a(n) = A182097(n) - A182097(n-1). - _R. J. Mathar_, Jan 27 2018
%p A078027 seq(coeff(series((1-x)/(1-x^2-x^3), x, n+1), x, n), n = 0..60); # _G. C. Greubel_, Aug 04 2019
%t A078027 CoefficientList[Series[(1-x)/(1-x^2-x^3), {x,0,60}], x] (* _G. C. Greubel_, Aug 04 2019 *)
%t A078027 LinearRecurrence[{0,1,1},{1,-1,1},60] (* _Harvey P. Dale_, Jun 20 2020 *)
%o A078027 (PARI) Vec((1-x)/(1-x^2-x^3)+O(x^60)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o A078027 (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x)/(1-x^2-x^3) )); // _G. C. Greubel_, Aug 04 2019
%o A078027 (Sage) ((1-x)/(1-x^2-x^3)).series(x, 60).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 04 2019
%o A078027 (GAP) a:=[1,-1,1];; for n in [4..60] do a[n]:=a[n-2]+a[n-3]; od; a; # _G. C. Greubel_, Aug 04 2019
%Y A078027 The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.
%K A078027 sign,easy
%O A078027 0,11
%A A078027 _N. J. A. Sloane_, Nov 17 2002