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

2, 3, 11, 26, 71, 183, 482, 1259, 3299, 8634, 22607, 59183, 154946, 405651, 1062011, 2780378, 7279127, 19056999, 49891874, 130618619, 341963987, 895273338, 2343856031, 6136294751, 16065028226, 42058789923, 110111341547

Offset: 0

Author

Wolfdieter Lang, Nov 01 2014

Keywords

• nonn
• easy

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.

Crossrefs

Cf. A000045, A059929, A079472, A121801, A248156.

Programs

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

Mathematica

CoefficientList[Series[(2 - x + x^2)/((1 + x) (1 - 3 x + x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Nov 01 2014 *)
With[{F=Fibonacci}, Table[F[2*n+2] +F[n]*F[n+1] +(-1)^n, {n,0,40}]] (* G. C. Greubel, May 30 2025 *)

SageMath

def A248161(n): return fibonacci(2*n+2) +fibonacci(n)*fibonacci(n+1) +(-1)^n
print([A248161(n) for n in range(41)]) # G. C. Greubel, May 30 2025

Formula

a(n) = F(n+3)^2 - (F(n)^2 + F(n+1)^2 + F(n+2)^2), F(n) = A000045(n).

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

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.

From G. C. Greubel, May 30 2025: (Start)

a(n) = Fibonacci(2*n+2) + Fibonacci(n)*Fibonacci(n+1) + (-1)^n.

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)

Extensions

Typo in formula fixed by Vincenzo Librandi, Nov 01 2014