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 A120893 #38 May 31 2025 05:19:04
%S A120893 1,1,5,17,65,241,901,3361,12545,46817,174725,652081,2433601,9082321,
%T A120893 33895685,126500417,472105985,1761923521,6575588101,24540428881,
%U A120893 91586127425,341804080817,1275630195845,4760716702561,17767236614401
%N A120893 a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3); a(0)=1, a(1)=1, a(2)=5.
%C A120893 For n>1, hypotenuse of primitive Pythagorean triangles having an angle nearing Pi/3 for larger values of sides. Complete triple (X,Y,Z),X<Y<Z is given by X=A120892(n),Y=A001353(n),Z=a(n) with recurrence relations X(i+1)=2*{a(i)-(-1)^i}-X(i-1) ; Y(i+1)=2*T(i)-T(i-1)-(-1)^i, where T(i)=Y(i)+a(i)] a(n)=2*A120892(n)-(-1)^n.
%H A120893 Vincenzo Librandi, <a href="/A120893/b120893.txt">Table of n, a(n) for n = 0..1000</a>
%H A120893 R. C. Alperin, <a href="https://www.fq.math.ca/Papers/57-4/alperin07132019.pdf">A family of nonlinear recurrences and their linear solutions</a>, Fib. Q., 57:4 (2019), 318-321.
%H A120893 W. K. Alt, <a href="https://wyattalt.com/files/thesis.pdf">Enumeration of Domino Tilings on the Projective Grid Graph</a>, A Thesis Presented to The Division of Mathematics and Natural Sciences, Reed College, May 2013.
%H A120893 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-1).
%F A120893 Union of A103772 and A103974. a(n)=2*{2*a(n-1) + (-1)^n} - a(n-2) ; a(0)=1,a(1)=1.
%F A120893 a(n) = [(-1)^n+(2-sqrt(3))^n+(2+sqrt(3))^n]/3. - _Emeric Deutsch_, Jul 24 2006
%F A120893 O.g.f: -(-1+2*x+x^2)/((1+x)*(x^2-4*x+1)). - _R. J. Mathar_, Dec 02 2007
%F A120893 a(n)+a(n+1) = A003699(n+1), n>0. - _R. J. Mathar_, Oct 15 2013
%p A120893 a[0]:=1: a[1]:=1: a[2]:=5: for n from 3 to 40 do a[n]:=3*a[n-1]+3*a[n-2]-a[n-3] od: seq(a[n],n=0..30); # _Emeric Deutsch_, Jul 24 2006
%t A120893 Transpose[NestList[Flatten[{Rest[#],3Last[#]+3#[[2]]- First[#]}]&, {1,1,5},25]][[1]] (* or *)
%t A120893 CoefficientList[Series[(1-2 x-x^2)/(1-3 x-3 x^2+x^3),{x,0,25}],x] (* _Harvey P. Dale_, Mar 27 2011 *)
%o A120893 (Magma) [Floor(((-1)^n+(2-Sqrt(3))^n+(2+Sqrt(3))^n)/3): n in [0..40]]; // _Vincenzo Librandi_, Jul 09 2012
%K A120893 nonn,easy
%O A120893 0,3
%A A120893 _Lekraj Beedassy_, Jul 14 2006
%E A120893 More terms from _Emeric Deutsch_, Jul 24 2006