A088137 Generalized Gaussian Fibonacci integers.

0, 1, 2, 1, -4, -11, -10, 13, 56, 73, -22, -263, -460, -131, 1118, 2629, 1904, -4079, -13870, -15503, 10604, 67717, 103622, 4093, -302680, -617639, -327238, 1198441, 3378596, 3161869, -3812050, -17109707, -22783264, 5762593, 79874978, 142462177, 45299420, -336787691

Offset

0,3

Comments

The Lucas U(P=2, Q=3) sequence. - R. J. Mathar, Oct 24 2012

Hence for n >= 0, a(n+2)/a(n+1) equals the continued fraction 2 - 3/(2 - 3/(2 - 3/(2 - ... - 3/2))) with n 3's. - Greg Dresden, Oct 06 2019

With different signs, 0, 1, -2, 1, 4, -11, 10, 13, -56, 73, 22, -263, 460, ... also the Lucas U(-2,3) sequence. - R. J. Mathar, Jan 08 2013

From Peter Bala, Apr 01 2018: (Start)

The companion Lucas sequence V(n,2,3) is A087455.

Define a binary operation o on rational numbers by x o y = (x + y)/(1 - 2*x*y). This is a commutative and associative operation with identity 0. Then 1 o 1 o ... o 1 (n terms) = a(n)/A087455(n). Cf. A025172 and A127357. (End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.

Ronald Orozco López, Deformed Differential Calculus on Generalized Fibonacci Polynomials, arXiv:2211.04450 [math.CO], 2022.

Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 16.

Wikipedia, Lucas sequence

Index entries for linear recurrences with constant coefficients, signature (2,-3).

Index entries for Lucas sequences

Formula

a(n) = 3^(n/2)*sin(n*atan(sqrt(2)))/sqrt(2).

|3*A087455(n) - A087455(n+1)| = 2*a(n+1) or 3*A087455(n) + A087455(n+1) = 2*a(n+1). - Creighton Dement, Aug 02 2004

G.f.: x/(1 - 2*x + 3*x^2).

E.g.f.: exp(x)*sin(sqrt(2)*x)/sqrt(2).

a(n) = 2*a(n-1) - 3*a(n-2) for n > 1, a(0)=0, a(1)=1.

a(n) = ((1 + i*sqrt(2))^n - (1 - i*sqrt(2))^n)/(2*i*sqrt(2)), where i=sqrt(-1).

a(n) = Im((1 + i*sqrt(2))^n/sqrt(2)).

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k+1)(-2)^k.

3^(n+1) = 9*(A087455(n))^2 + 2*(A087455(n+1))^2 - 2*(a(n+2))^2; 3^n = a(n+1)^2 + 3*a(n)^2 - 2*a(n+1)*a(n) for n > 0 - Creighton Dement, Jan 20 2005

G.f.: G(0)*x/(2*(1-x)), where G(k) = 1 + 1/(1 - x*(2*k+1)/(x*(2*k+3) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013

G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 - 3*x)/( x*(4*k+4 - 3*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 06 2013

a(n+1) = Sum_{k=0..n} A123562(n,k). - Philippe Deléham, Nov 23 2013

a(n) = n*hypergeom([(1-n)/2,(2-n)/2],[3/2],-2). - Gerry Martens, Sep 03 2023

Maple

A[0]:= 0: A[1]:= 1:
for n from 2 to 100 do A[n]:= 2*A[n-1] - 3*A[n-2] od:
seq(A[n], n=0..100); # Robert Israel, Aug 05 2014

Mathematica

LinearRecurrence[{2, -3}, {0, 1}, 40] (* Harvey P. Dale, Nov 03 2014 *)

Prog

(Sage) [lucas_number1(n, 2, 3) for n in range(0, 38)] # Zerinvary Lajos, Apr 23 2009
(PARI) x='x+O('x^50); concat([0], Vec(x/(1-2*x+3*x^2))) \\ G. C. Greubel, Oct 22 2018
(Magma) [n le 2 select n-1 else 2*Self(n-1)-3*Self(n-2): n in [1..50]]; // G. C. Greubel, Oct 22 2018

Crossrefs

Cf. A084102, A088138, A045873, A088139.

Cf. A087455, A025172, A123562, A127357.

Sequence in context: A134028 A111479 A323937 * A205870 A371062 A354116

Adjacent sequences: A088134 A088135 A088136 * A088138 A088139 A088140

Keyword

sign,easy,changed

Author

Paul Barry, Sep 20 2003

Status

approved