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A061084 Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1). 19

%I

%S 1,2,-1,3,-4,7,-11,18,-29,47,-76,123,-199,322,-521,843,-1364,2207,

%T -3571,5778,-9349,15127,-24476,39603,-64079,103682,-167761,271443,

%U -439204,710647,-1149851,1860498,-3010349,4870847,-7881196,12752043,-20633239,33385282,-54018521

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

%C 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

%C 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

%C 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

%H Indranil Ghosh, <a href="/A061084/b061084.txt">Table of n, a(n) for n = 0..4771</a> (terms 0..500 from T. D. Noe)

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Q. Wang, <a href="https://www.semanticscholar.org/paper/On-generalized-Lucas-sequences-Wang-Akbari/7e33b3b79703dc6790fca133e8c92cc0cafcfe4a">On generalized Lucas sequences</a>, Contemp. Math. 531 (2010) 127-141, Table 2 (k=2).

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,1).

%H <a href="/index/Lu#Lucas">Index entries for Lucas sequences</a>

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

%F O.g.f.: (3*x+1)/(1+x-x^2). - _Len Smiley_, Dec 02 2001

%F a(n) = A039834(n+1)+3*A039834(n). - _R. J. Mathar_, Oct 30 2015

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

%t LinearRecurrence[{-1,1},{1,2},40] (* _Harvey P. Dale_, Nov 22 2011 *)

%o (Haskell)

%o a061084 n = a061084_list !! n

%o a061084_list = 1 : 2 : zipWith (-) a061084_list (tail a061084_list)

%o -- _Reinhard Zumkeller_, Feb 01 2014

%o (PARI) a(n)=([0,1; 1,-1]^n*[1;2])[1,1] \\ _Charles R Greathouse IV_, Feb 09 2017

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

%K sign,easy,nice

%O 0,2

%A Ulrich Schimke (ulrschimke(AT)aol.com)

%E Corrected by _T. D. Noe_, Oct 25 2006

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Last modified October 16 04:02 EDT 2018. Contains 316259 sequences. (Running on oeis4.)