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
Examples
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.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Crossrefs
Cf. A005132.
Programs
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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 -
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 *) -
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
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
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