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A088137 Generalized Gaussian Fibonacci integers. 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

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

LINKS

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

C. Dement, The Math Forum.

Wikipedia, Lucas sequence

Index entries for sequences related to 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

a(n+1) = tes(x^n) = -les(x^n)/3 x= 2('i) - 'k - 'jj' - 'ji' - 'jk' - 1. - Creighton Dement, Aug 02 2004

G.f.: x/(1-2x+3x^2); E.g.f.: exp(x)sin(sqrt(2)x)/sqrt(2); a(n)=2a(n-1)-3a(n-2), a(0)=0, a(1)=1; a(n)=((1+i*sqrt(2))^n-(1-i*sqrt(2))^n)/(2i*sqrt(2)); a(n)=Im{(1+i*sqrt(2))^n/sqrt(2)}; a(n)=sum{k=0..floor(n/2), C(n, 2k+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), 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) = 2*a(n-1)-3*a(n-2), with a(0) = 0, a(1) = 1. - Richard R. Forberg, Aug 05 2014

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

Join[{a=0, b=1}, Table[c=2*b-3*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)

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

PROG

(Sage) [lucas_number1(n, 2, 3) for n in xrange(0, 38)] # Zerinvary Lajos, Apr 23 2009

CROSSREFS

Cf. A084102, A088138, A045873, A088139.

Cf. A087455.

Sequence in context: A016544 A134028 A111479 * A205870 A064297 A052661

Adjacent sequences:  A088134 A088135 A088136 * A088138 A088139 A088140

KEYWORD

easy,sign

AUTHOR

Paul Barry, Sep 20 2003

STATUS

approved

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Last modified December 18 17:57 EST 2014. Contains 252173 sequences.