A061084 Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).

1, 2, -1, 3, -4, 7, -11, 18, -29, 47, -76, 123, -199, 322, -521, 843, -1364, 2207, -3571, 5778, -9349, 15127, -24476, 39603, -64079, 103682, -167761, 271443, -439204, 710647, -1149851, 1860498, -3010349, 4870847, -7881196, 12752043, -20633239, 33385282, -54018521

Offset

0,2

Comments

If we drop 1 and start with 2 this is the Lucas sequence V(-1,-1). G.f.: (2+x)/(1+x-x^2). In this case a(n) is also the trace of A^(-n), where A is the Fibomatrix ((1,1), (1,0)). - Mario Catalani (mario.catalani(AT)unito.it), Aug 17 2002

The positive sequence with g.f. (1+x-2x^2)/(1-x-x^2) gives the diagonal sums of the Riordan array (1+2x,x/(1-x)). - Paul Barry, Jul 18 2005

Pisano period lengths: 1, 3, 8, 6, 4, 24, 16, 12, 24, 12, 10, 24, 28, 48, 8, 24, 36, 24, 18, 12, .... (is this A106291?). - R. J. Mathar, Aug 10 2012

Links

Indranil Ghosh, Table of n, a(n) for n = 0..4771 (terms 0..500 from T. D. Noe)

Tanya Khovanova, Recursive Sequences

Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).

Q. Wang, On generalized Lucas sequences, Contemp. Math. 531 (2010) 127-141, Table 2 (k=2).

Wikipedia, Lucas sequence

Index entries for linear recurrences with constant coefficients, signature (-1,1).

Index entries for Lucas sequences

Formula

a(n) = (-1)^(n-1) * A000204(n-1).

O.g.f.: (3*x+1)/(1+x-x^2). - Len Smiley, Dec 02 2001

a(n) = A039834(n+1)+3*A039834(n). - R. J. Mathar, Oct 30 2015

Example

a(6) = a(4)-a(5) = -4 - 7 = -11.

Mathematica

LinearRecurrence[{-1, 1}, {1, 2}, 40] (* Harvey P. Dale, Nov 22 2011 *)

Prog

(Haskell)
a061084 n = a061084_list !! n
a061084_list = 1 : 2 : zipWith (-) a061084_list (tail a061084_list)
-- Reinhard Zumkeller, Feb 01 2014
(PARI) a(n)=([0, 1; 1, -1]^n*[1; 2])[1, 1] \\ Charles R Greathouse IV, Feb 09 2017

Crossrefs

Cf. A061083 for division, A000301 for multiplication and A000045 for addition - the common Fibonacci numbers.

Sequence in context: A160191 A268613 A268615 * A000032 A329723 A267551

Adjacent sequences: A061081 A061082 A061083 * A061085 A061086 A061087

Keyword

sign,easy,nice

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Extensions

Corrected by T. D. Noe, Oct 25 2006

Status

approved