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A248161 Expansion of (2-x+x^2)/((1+x)*(1-3*x+x^2)). 4

%I #14 Sep 08 2022 08:46:10

%S 2,3,11,26,71,183,482,1259,3299,8634,22607,59183,154946,405651,

%T 1062011,2780378,7279127,19056999,49891874,130618619,341963987,

%U 895273338,2343856031,6136294751,16065028226,42058789923,110111341547

%N Expansion of (2-x+x^2)/((1+x)*(1-3*x+x^2)).

%C 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.

%D R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.

%F a(n) = -(F(n)^2 + F(n+1)^2 + F(n+2)^2 - F(n+3)^2).

%F a(n) = (4*(-1)^n + F(2*n) +6*F(2*n+2))/5, with the Fibonacci numbers F = A000045.

%F 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.

%t CoefficientList[Series[(2 - x + x^2)/((1 + x) (1 - 3 x + x^2)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Nov 01 2014 *)

%o (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

%Y Cf. A000045, A059929, A079472, A121801.

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, Nov 01 2014

%E Typo in formula fixed by _Vincenzo Librandi_, Nov 01 2014

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)