A079053 Recamán Fibonacci variation: a(1)=1; a(2)=2; for n > 2, a(n) = a(n-1)+a(n-2)-F(n) if that number is positive and not already in the sequence, otherwise a(n) = a(n-1)+a(n-2)+F(n) where F(n) denotes the n-th Fibonacci number.

1, 2, 5, 4, 14, 10, 11, 42, 19, 6, 114, 264, 145, 32, 787, 1806, 996, 218, 5395, 12378, 6827, 1494, 36978, 84840, 46793, 10240, 253451, 581502, 320724, 70186, 1737179, 3985674, 2198275, 481062, 11906802, 27318216, 15067201, 3297248, 81610435

Offset

1,2

Comments

Starting with other initial values a(1)=x a(2)=y gives the same kind of recurrence relations.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Formula

For n>2, if n==0 or 2 (mod 4) a(n)=2*a(n-1)-a(n-2)-a(n-4); if n==1 or 3 (mod 4) a(n)=a(n-2)+2*a(n-3)+a(n-4) lim n ->infinity a(4n)/a(4n-1)=2.29433696806047607330083539....; lim n ->infinity a(4n-1)/a(4n-2)=24.7510757456062014116731647..; lim n ->infinity a(4n-2)/a(4n-3)=0.218836132868832627648170038...; lim n ->infinity a(4n-3)/a(4n-4)=0.551544105222898180785441647...

Empirical g.f.: x*(26*x^10+68*x^8+6*x^7-4*x^6+16*x^5-12*x^4-5*x^2-x-1) / ((x^2+x-1)*(x^4+3*x^2+1)). - Colin Barker, Jun 26 2013

Example

a(10)=6 because a(9)+a(8)-F(10)=19+42-55=6 and 6 is not already in the sequence. a(11)=42 because a(10)+a(9)-F(11)=6+19-89 < 0 then a(11)=6+19+89=114.

Mathematica

a[1] = 1; a[2] = 2; a[n_] := a[n] = (an = a[n-1] + a[n-2] - Fibonacci[n]; If[an > 0 && ! MemberQ[Array[a, n-1], an], an, a[n-1] + a[n-2] + Fibonacci[n]]); Table[a[n], {n, 1, 39}] (* Jean-François Alcover, Jun 18 2012 *)

Prog

(PARI) m=200; a=vector(m); a[1]=1; a[2]=2; for(n=3, m, a[n]=if(n<0, 0, if(abs(sign(a[n-1]+a[n-2]-fibonacci(n))-1)+setsearch(Set(vector(n-1, i, a[i])), a[n-1]+a[n-2]-fibonacci(n)), a[n-1]+a[n-2]+fibonacci(n), a[n-1]+a[n-2]-fibonacci(n)))); a - corrected by Colin Barker, Jun 26 2013
(Haskell)
import Data.Set (Set, fromList, notMember, insert)
a079053 n = a079053_list !! (n-1)
a079053_list = 1 : 2 : r (fromList [1, 2]) 1 1 1 2 where
  r :: Set Integer -> Integer -> Integer -> Integer -> Integer -> [Integer]
  r s i j x y = if v > 0 && v `notMember` s
                   then v : r (insert v s) j fib y v
                   else w : r (insert w s) j fib y w where
    fib = i + j
    v = x + y - fib
    w = x + y + fib
for_bFile = take 1000 a079053_list -- Reinhard Zumkeller, Mar 14 2011

Crossrefs

Cf. A005132.

Sequence in context: A102468 A252668 A225046 * A224272 A189942 A251554

Adjacent sequences: A079050 A079051 A079052 * A079054 A079055 A079056

Keyword

nice,nonn

Author

Benoit Cloitre, Feb 02 2003

Status

approved