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

0,4

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 Quart. 13 (1975), 233-239. - Ron Knott, Oct 29 2010

Ali Reza Ashrafi, Jernej Azarija, Khadijeh Fathalikhani, Sandi Klavžar, and Marko Petkovšek, Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.

Paul Barry, Pascal-like Sprugnoli arrays, arXiv:2606.22070 [math.CO], 2026. See p. 7.

V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart. 13(4) (1975), 350-356.

M. A. Nyblom, Counting Palindromic Binary Strings Without r-Runs of Ones, J. Int. Seq. 16 (2013) Art. 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).

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

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 *)
(* Alternative: *)
LinearRecurrence[{0, 1, 0, 1}, {0, 1, 1, 2}, 60] (* Harvey P. Dale, Nov 07 2016 *)
(* Alternative: *)
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 *)

Prog

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

Crossrefs

Cf. A000045, A001622, A051792, A079977.

Sequence in context: A262090 A239881 A051792 * A272912 A123231 A246995

Adjacent sequences: A053599 A053600 A053601 * A053603 A053604 A053605

Keyword

nonn,easy,changed

Author

Michael Somos, Jan 17 2000

Status

approved