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A007538
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A self-generating sequence: there are a(n) 3's between successive 2's.
(Formerly M0432)
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13
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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
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OFFSET
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1,1
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COMMENTS
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(a(n)) is the unique fixed point of the morphism 2->233, 3->2333 (immediate from its definition). - Michel Dekking, Feb 21 2017
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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.
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Sequence in context: A102313 A262955 A276859 * A242285 A238756 A025076
Adjacent sequences: A007535 A007536 A007537 * A007539 A007540 A007541
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, Simon Plouffe
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STATUS
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approved
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