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 A078010 #46 Sep 01 2023 04:23:52
%S A078010 1,0,1,3,4,9,19,36,73,147,292,585,1171,2340,4681,9363,18724,37449,
%T A078010 74899,149796,299593,599187,1198372,2396745,4793491,9586980,19173961,
%U A078010 38347923,76695844,153391689,306783379,613566756,1227133513,2454267027,4908534052
%N A078010 Expansion of (1-x)/(1 - x - x^2 - 2*x^3).
%C A078010 For n > 0, a(n) is the number of ways to tile a strip of length n with squares, dominoes, and two colors of trominoes, with the restriction that the first tile cannot be a square. - _Greg Dresden_ and Bora Bursalı, Aug 31 2023
%H A078010 G. C. Greubel, <a href="/A078010/b078010.txt">Table of n, a(n) for n = 0..1000</a>
%H A078010 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,2).
%F A078010 a(0)=1, a(1)=0, a(2)=1, a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n > 2. - _Philippe Deléham_, Sep 19 2006
%F A078010 a(n) + a(n+1) = A122552(n+1). - _Philippe Deléham_, Sep 25 2006
%F A078010 If p[1]=0, p[2]=1, p[i]=3, (i>2), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n >= 1, a(n)=det A. - _Milan Janjic_, May 02 2010
%F A078010 For n > 3, a(n) = A077947(n-2) + 2*A077947(n-3), with A077947 beginning (1, 2, 5, 9, 18, 37, ...); "1" has offset 1. - _Gary W. Adamson_, May 13 2013
%F A078010 a(n) = 2^(n-1) - 3*floor((2^(n-1))/7) - 1, for n >= 1. - _Ridouane Oudra_, Dec 02 2019
%F A078010 G.f.: (1 - x) / ((1 - 2*x) * (1 + x + x^2)). - _Michael Somos_, Nov 18 2020
%e A078010 a(6) = 19 = A077947(4) + 2*A077947(3) = 9 + 2*5 = 19.
%e A078010 G.f. = 1 + x^2 + 3*x^3 + 4*x^4 + 9*x^5 + 19*x^6 + 36*x^7 + 73*x^8 + ... - _Michael Somos_, Nov 18 2020
%t A078010 CoefficientList[Series[(1-x)/(1-x-x^2-2*x^3), {x,0,50}],x] (* _Harvey P. Dale_, Mar 17 2011 *)
%t A078010 LinearRecurrence[{1, 1, 2}, {1, 0, 1}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 24 2012 *)
%o A078010 (PARI) Vec((1-x)/(1-x-x^2-2*x^3)+O(x^50)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o A078010 (PARI) {a(n) = ([0, 1, 1; 1, 1, 0; 0, 2, 0]^n)[1, 1]}; /* _Michael Somos_, Nov 18 2020 */
%o A078010 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x-x^2-2*x^3) )); // _G. C. Greubel_, Jun 28 2019
%o A078010 (Sage) ((1-x)/(1-x-x^2-2*x^3)).series(x, 50).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 28 2019
%o A078010 (GAP) a:=[1,0,1];; for n in [4..50] do a[n]:=a[n-1]+a[n-2]+2*a[n-3]; od; a; # _G. C. Greubel_, Jun 28 2019
%Y A078010 Cf. A033138, A077947.
%K A078010 nonn,easy
%O A078010 0,4
%A A078010 _N. J. A. Sloane_, Nov 17 2002