A036299 Binary Fibonacci (or rabbit) sequence.

1, 10, 101, 10110, 10110101, 1011010110110, 101101011011010110101, 1011010110110101101011011010110110, 1011010110110101101011011010110110101101011011010110101

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• base

Comments

A055642(a(n)) = A000045(n+2). - Reinhard Zumkeller, Jul 06 2014

References

• N. G. De Bruijn, (1989, January). Updown generation of Beatty sequences. Koninklijke Nederlandsche Akademie van Wetenschappen (Indationes Math.), Proc., Ser. A, 92:4 (1968), 385-407. See Fig. 3.
• J. Kappraff, D. Blackmore and G. Adamson, Phyllotaxis as a dynamical system: a study in number, Chap. 17 of Jean and Barabe, eds., Symmetry in Plants, World Scientific, Studies in Math. Biology and Medicine, Vol. 4.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..14
• M. S. El Naschie, Statistical geometry of a Cantor discretum and semiconductors, Computers Math. Applic., 29 (No, 12, 1995), 103-110.
• C. J. Glasby, S. P. Glasby and F. Pleijel, Worms by number, Proc. Roy. Soc. B, Proc. Biol. Sci. 275 (1647) (2008) 2071-2076.
• H. W. Gould, J. B. Kim and V. E. Hoggatt, Jr., Sequences associated with t-ary coding of Fibonacci's rabbits, Fib. Quart., 15 (1977), 311-318.

Crossrefs

Cf. A005614, A003849.

Column k=10 of A144287.

Programs

Haskell

a036299 n = a036299_list !! n
a036299_list = map read rabbits :: [Integer] where
   rabbits = "1" : "10" : zipWith (++) (tail rabbits) rabbits
-- Reinhard Zumkeller, Jul 06 2014

Mathematica

nxt[{a_,b_}]:=FromDigits[Join[IntegerDigits[b],IntegerDigits[a]]]; Transpose[NestList[{Last[#],nxt[#]}&,{1,10},10]][[1]] (* Harvey P. Dale, Oct 16 2011 *)

Python

def aupton(terms):
  alst = [1, 10]
  while len(alst) < terms: alst.append(int(str(alst[-1]) + str(alst[-2])))
  return alst[:terms]
print(aupton(9)) # Michael S. Branicky, Jan 10 2021

Formula

a(n+1) = concatenation of a(n) and a(n-1).