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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).

%I A061084 #50 Jun 14 2025 04:21:00
%S A061084 1,2,-1,3,-4,7,-11,18,-29,47,-76,123,-199,322,-521,843,-1364,2207,
%T A061084 -3571,5778,-9349,15127,-24476,39603,-64079,103682,-167761,271443,
%U A061084 -439204,710647,-1149851,1860498,-3010349,4870847,-7881196,12752043,-20633239,33385282,-54018521
%N A061084 Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2) - a(n-1).
%C A061084 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 A061084 The positive sequence with g.f. (1+x-2*x^2)/(1-x-x^2) gives the diagonal sums of the Riordan array (1+2*x, x/(1-x)). - _Paul Barry_, Jul 18 2005
%C A061084 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 A061084 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 A061084 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A061084 Kai Wang, <a href="https://www.researchgate.net/publication/337943524_Fibonacci_Numbers_And_Trigonometric_Functions_Outline">Fibonacci Numbers And Trigonometric Functions Outline</a>, (2019).
%H A061084 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 A061084 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>
%H A061084 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,1).
%H A061084 <a href="/index/Lu#Lucas">Index entries for Lucas sequences</a>
%F A061084 a(n) = (-1)^(n-1) * A000204(n-1).
%F A061084 O.g.f.: (1+3*x)/(1+x-x^2). - _Len Smiley_, Dec 02 2001
%F A061084 a(n) = A039834(n+1) + 3*A039834(n). - _R. J. Mathar_, Oct 30 2015
%F A061084 From _G. C. Greubel_, Jun 14 2025: (Start)
%F A061084 a(n) = A000032(1-n).
%F A061084 E.g.f.: exp(-x/2)*( cosh(p*x) + sqrt(5)*sinh(p*x) ), where p = sqrt(5)/2. (End)
%e A061084 a(6) = a(4)-a(5) = -4 - 7 = -11.
%t A061084 LinearRecurrence[{-1,1},{1,2},40] (* _Harvey P. Dale_, Nov 22 2011 *)
%t A061084 LucasL[1-Range[0, 40]] (* _G. C. Greubel_, Jun 14 2025 *)
%o A061084 (Haskell)
%o A061084 a061084 n = a061084_list !! n
%o A061084 a061084_list = 1 : 2 : zipWith (-) a061084_list (tail a061084_list)
%o A061084 -- _Reinhard Zumkeller_, Feb 01 2014
%o A061084 (PARI) a(n)=([0,1; 1,-1]^n*[1;2])[1,1] \\ _Charles R Greathouse IV_, Feb 09 2017
%o A061084 (Magma)
%o A061084 A061084:= func< n | Lucas(1-n) >; // _G. C. Greubel_, Jun 14 2025
%o A061084 (SageMath)
%o A061084 def A061084(n): return lucas_number2(1-n,1,-1) # _G. C. Greubel_, Jun 14 2025
%Y A061084 Cf. A061083 for division, A000301 for multiplication and A000045 for addition - the common Fibonacci numbers.
%K A061084 sign,easy,nice
%O A061084 0,2
%A A061084 Ulrich Schimke (ulrschimke(AT)aol.com)
%E A061084 Corrected by _T. D. Noe_, Oct 25 2006