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 A078042 #40 May 03 2023 15:17:01
%S A078042 1,-2,3,-6,11,-20,37,-68,125,-230,423,-778,1431,-2632,4841,-8904,
%T A078042 16377,-30122,55403,-101902,187427,-344732,634061,-1166220,2145013,
%U A078042 -3945294,7256527,-13346834,24548655,-45152016,83047505,-152748176,280947697,-516743378,950439251,-1748130326,3215312955
%N A078042 Expansion of (1-x)/(1+x-x^2+x^3).
%C A078042 Absolute values give coordination sequence for (3,infinity,infinity) tiling of hyperbolic plane. - _N. J. A. Sloane_, Dec 29 2015
%C A078042 a(n) is the upper left entry of the n-th power of the 3 X 3 matrix M = [-2, -2, 1; 1, 1, 0; 1, 0, 0]; a(n) = M^n [1, 1]. - _Philippe Deléham_, Apr 19 2023
%H A078042 G. C. Greubel, <a href="/A078042/b078042.txt">Table of n, a(n) for n = 0..1000</a>
%H A078042 J. W. Cannon and P. Wagreich, <a href="http://dx.doi.org/10.1007/BF01444714">Growth functions of surface groups</a>, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
%H A078042 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,1,-1).
%F A078042 a(n) = -a(n-1) + a(n-2) - a(n-3) for n > 2; a(0)=1, a(1)=-2, a(2)=3. - _Harvey P. Dale_, Jun 01 2012
%F A078042 a(n) = (-1)^n * A001590(n+2).
%F A078042 a(n) = Sum_{k=0..n} A188316(n,k)*(-2)^k. - _Philippe Deléham_, Apr 19 2023
%t A078042 CoefficientList[Series[(1-x)/(1+x-x^2+x^3),{x,0,40}],x] (* or *) LinearRecurrence[{-1,1,-1},{1,-2,3},40] (* _Harvey P. Dale_, Jun 01 2012 *)
%o A078042 (PARI) Vec((1-x)/(1+x-x^2+x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o A078042 (Magma) [n le 3 select -n*(-1)^n else -Self(n-1)+Self(n-2)-Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Dec 30 2015
%Y A078042 Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.
%K A078042 sign,easy
%O A078042 0,2
%A A078042 _N. J. A. Sloane_, Nov 17 2002