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 A053602 a(n) = a(n-1) - (-1)^n*a(n-2), a(0)=0, a(1)=1. 13
 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 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 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 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 *) a={0, 1}; Do[AppendTo[a, a[[-1]]-(-1)^(Length[a])a[[-2]]], {49}]; a (* Waldemar Puszkarz, 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 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 March 24 04:20 EDT 2018. Contains 301177 sequences. (Running on oeis4.)