A053602 a(n) = a(n-1) - (-1)^n*a(n-2), a(0)=0, a(1)=1.

0, 1, 1, 2, 1, 3, 2, 5, 3, 8, 5, 13, 8, 21, 13, 34, 21, 55, 34, 89, 55, 144, 89, 233, 144, 377, 233, 610, 377, 987, 610, 1597, 987, 2584, 1597, 4181, 2584, 6765, 4181, 10946, 6765, 17711, 10946, 28657, 17711, 46368, 28657, 75025, 46368, 121393, 75025

Offset: 0

Author

Michael Somos, Jan 17 2000

Keywords

• nonn
• easy

Comments

If b(0)=0, b(1)=1 and b(n) = b(n-1) + (-1)^n*b(n-2), then a(n) = b(n+3). - Jaume Oliver Lafont, Oct 03 2009

a(n) is the number of palindromic compositions of n-1 into parts of 1 and 2. a(7) = 5 because we have 2+2+2, 2+1+1+2, 1+2+2+1, 1+1+2+1+1, 1+1+1+1+1+1. - Geoffrey Critzer, Mar 17 2014

a(n) is the number of palindromic compositions of n into odd parts (the corresponding generating function follows easily from Theorem 1.2 of the Hoggatt et al. reference). Example: a(7) = 5 because we have 7, 1+5+1, 3+1+3, 1+1+3+1+1, 1+1+1+1+1+1+1. - Emeric Deutsch, Aug 16 2016

The ratio of a(n)/a(n-1) oscillates between phi-1 and phi+1 as n tends to infinity, where phi is golden ratio (A001622). - Waldemar Puszkarz, Oct 10 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239. - _Ron Knott_, Oct 29 2010
• A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.
• V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
• M. A. Nyblom, Counting Palindromic Binary Strings Without r-Runs of Ones, J. Int. Seq. 16 (2013) #13.8.7, P_2(n)
• Index entries for two-way infinite sequences
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1)

Crossrefs

Cf. A000045, A001622, A051792, A079977.

Programs

Magma

I:=[0,1,1,2]; [n le 4 select I[n] else Self(n-2)+Self(n-4): n in [1..50]]; // Vincenzo Librandi Oct 10 2017

Maple

a[0] := 0: a[1] := 1: for n from 2 to 60 do a[n] := a[n-1]-(-1)^n*a[n-2] end do: seq(a[n], n = 0 .. 50); # Emeric Deutsch, Oct 09 2017

Mathematica

nn=50;CoefficientList[Series[x (1+x+x^2)/(1-x^2-x^4),{x,0,nn}],x] (* Geoffrey Critzer, Mar 17 2014 *)
LinearRecurrence[{0,1,0,1},{0,1,1,2},60] (* Harvey P. Dale, Nov 07 2016 *)
RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]-(-1)^n a[n-2]}, a, {n, 50}] (* Vincenzo Librandi, Oct 10 2017 *)

PARI

a(n)=fibonacci(n\2+n%2*2)

SageMath

[fibonacci(n//2 + 2*(n%2)) for n in range(61)] # G. C. Greubel, Dec 06 2022

Formula

G.f.: x*(1 + x + x^2)/(1 - x^2 - x^4).

a(n) = a(n-2) + a(n-4).

a(2n) = F(n), a(2n-1) = F(n+1) where F() is Fibonacci sequence.

a(3-n) = A051792(n).

a(3)=1, a(4)=2, a(n+2) = a(n+1) + sign(a(n) - a(n+1))*a(n), n > 4. - Benoit Cloitre, Apr 08 2002

a(n) = A079977(n-1) + A079977(n-2) + A079977(n-3), n > 2. - Ralf Stephan, Apr 26 2003

a(0) = 0, a(1) = 1; a(2n) = a(2n-1) - a(2n-2); a(2n+1) = a(2n) + a(2n-1). - Amarnath Murthy, Jul 21 2005