login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A088137 Generalized Gaussian Fibonacci integers. 13
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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 3 16:24 EST 2016. Contains 278745 sequences.