A007538 A self-generating sequence: there are a(n) 3's between successive 2's.

2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy
• nice

Comments

(a(n)) is the unique fixed point of the morphism 2->233, 3->2333 (immediate from its definition). - Michel Dekking, Feb 21 2017

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Bryce Emerson Blackham, Subtraction Games: Range and Strict Periodicity, Master's thesis, 2018.
• The Fifty-Fourth William Lowell Putnam Mathematical Competition, Problem A-6, Amer. Math. Monthly, 101 (1994), 727-728.
• The Fifty-Fourth William Lowell Putnam Mathematical Competition, Problem A-6, Math. Mag., 67 (No. 2, 1994), 157-158.

Programs

Haskell

a007538 n = f n 2 2 2 where
   f 1 b   = b
   f n b 0 i = f (n - 1) 2 (a007538 i) (i + 1)
   f n b c i = f (n - 1) 3 (c - 1) i
-- Reinhard Zumkeller, Feb 14 2012

Mathematica

f[n_, b_, c_, i_] := f[n, b, c, i] = If[n == 1, b, If[c == 0 , f[n-1, 2, a[i], i+1], f[n-1, 3, c-1, i]]]; a[n_] := f[n, 2, 2, 2]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 15 2013, after Reinhard Zumkeller *)
Table[Floor[n (1 + Sqrt@ 3)] - Floor[(n - 1) (1 + Sqrt@ 3)], {n, 120}] (* Michael De Vlieger, Oct 08 2016 *)
t = {2};Table[If[t[[i]] == 2, AppendTo[t, #] & /@ {3, 3, 2}, AppendTo[t, #] & /@ {3, 3, 3, 2}], {i, 20}];t   (* Horst H. Manninger, Jan 11 2024 *)

Formula

a(n) = floor( n*(1+sqrt(3)) ) - floor( (n-1)*(1+sqrt(3)) ).

a(n) = f(n,2,2,2) with f(n,b,c,i) = if n=1 then b else (if c=0 then f(n-1,2,a(i),i+1) else f(n-1,3,c-1,i)). - Reinhard Zumkeller, May 25 2009

a(n) = A080757(n-1) + 1; a(n) = A188068(n) + 2. - Reinhard Zumkeller, Feb 14 2012

a(A188069(n)) = 2; a(A188070(n)) = 3. - Reinhard Zumkeller, Feb 14 2012