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 A248161 #20 May 30 2025 19:29:25
%S A248161 2,3,11,26,71,183,482,1259,3299,8634,22607,59183,154946,405651,
%T A248161 1062011,2780378,7279127,19056999,49891874,130618619,341963987,
%U A248161 895273338,2343856031,6136294751,16065028226,42058789923,110111341547
%N A248161 Expansion of (2-x+x^2)/((1+x)*(1-3*x+x^2)).
%C A248161 The negative of this sequence provides the first component of the square of [F(n), F(n+1), F(n+2), F(n+3)], for n >= 0, where F(n) = A000045(n), in the Clifford algebra Cl_2 over Euclidean 2-space. The whole quartet of sequences for this square is [-a(n), A079472(n+1), A059929(n), A121801(n+1)]. See the Oct 15 2014 comment in A147973 where also a reference is given.
%H A248161 G. C. Greubel, <a href="/A248161/b248161.txt">Table of n, a(n) for n = 0..1000</a>
%H A248161 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.
%F A248161 a(n) = F(n+3)^2 - (F(n)^2 + F(n+1)^2 + F(n+2)^2), F(n) = A000045(n).
%F A248161 a(n) = (6*F(2*n+2) + F(2*n) + 4*(-1)^n)/5, with the Fibonacci numbers F = A000045.
%F A248161 O.g.f.: (2-x+x^2)/((1+x)*(1-3*x+x^2)) = (4/(1+x) + (x+6)/(1-3*x+x^2))/5.
%F A248161 From _G. C. Greubel_, May 30 2025: (Start)
%F A248161 a(n) = Fibonacci(2*n+2) + Fibonacci(n)*Fibonacci(n+1) + (-1)^n.
%F A248161 E.g.f.: (1/5)*(exp(3*x/2)*(6*cosh(sqrt(5)*x/2) + 4*sqrt(5)*sinh(sqrt(5)*x/2)) + 4*exp(-x)). (End)
%t A248161 CoefficientList[Series[(2 - x + x^2)/((1 + x) (1 - 3 x + x^2)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Nov 01 2014 *)
%t A248161 With[{F=Fibonacci}, Table[F[2*n+2] +F[n]*F[n+1] +(-1)^n, {n,0,40}]] (* _G. C. Greubel_, May 30 2025 *)
%o A248161 (Magma) [-(Fibonacci(n)^2 +Fibonacci(n+1)^2 + Fibonacci(n+2)^2 - Fibonacci(n+3)^2): n in [0..30]]; // _Vincenzo Librandi_, Nov 01 2014
%o A248161 (SageMath)
%o A248161 def A248161(n): return fibonacci(2*n+2) +fibonacci(n)*fibonacci(n+1) +(-1)^n
%o A248161 print([A248161(n) for n in range(41)]) # _G. C. Greubel_, May 30 2025
%Y A248161 Cf. A000045, A059929, A079472, A121801, A248156.
%K A248161 nonn,easy
%O A248161 0,1
%A A248161 _Wolfdieter Lang_, Nov 01 2014
%E A248161 Typo in formula fixed by _Vincenzo Librandi_, Nov 01 2014