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A053602 a(n) = a(n-1) - (-1)^n*a(n-2), a(0)=0, a(1)=1. 12
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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=0..50.

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.

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)=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

MATHEMATICA

nn=50; CoefficientList[Series[x (1+x+x^2)/(1-x^2-x^4), {x, 0, nn}], x] (* Geoffrey Critzer, Mar 17 2014 *)

PROG

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

CROSSREFS

a(3-n) = A051792(n). Cf. A000045.

Sequence in context: A239881 A051792 * A272912 A123231 A246995 A238782

Adjacent sequences:  A053599 A053600 A053601 * A053603 A053604 A053605

KEYWORD

nonn,easy

AUTHOR

Michael Somos, Jan 17 2000

STATUS

approved

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Last modified October 1 16:40 EDT 2016. Contains 276659 sequences.