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

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

• sign
• easy
• nice

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-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

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

Examples

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

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

Crossrefs

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

Programs

Haskell

a061084 n = a061084_list !! n
a061084_list = 1 : 2 : zipWith (-) a061084_list (tail a061084_list)
-- Reinhard Zumkeller, Feb 01 2014

Magma

A061084:= func< n | Lucas(1-n) >; // G. C. Greubel, Jun 14 2025

Mathematica

LinearRecurrence[{-1,1},{1,2},40] (* Harvey P. Dale, Nov 22 2011 *)
LucasL[1-Range[0, 40]] (* G. C. Greubel, Jun 14 2025 *)

PARI

a(n)=([0,1; 1,-1]^n*[1;2])[1,1] \\ Charles R Greathouse IV, Feb 09 2017

SageMath

def A061084(n): return lucas_number2(1-n,1,-1) # G. C. Greubel, Jun 14 2025

Formula

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

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

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

From G. C. Greubel, Jun 14 2025: (Start)

a(n) = A000032(1-n).

E.g.f.: exp(-x/2)*( cosh(p*x) + sqrt(5)*sinh(p*x) ), where p = sqrt(5)/2. (End)

Extensions

Corrected by T. D. Noe, Oct 25 2006