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A100982
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Number of admissible sequences of order j; related to 3x+1 problem and Wagon's constant.
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8
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1, 1, 2, 3, 7, 12, 30, 85, 173, 476, 961, 2652, 8045, 17637, 51033, 108950, 312455, 663535, 1900470, 5936673, 13472296, 39993895, 87986917, 257978502, 820236724, 1899474678, 5723030586, 12809477536, 38036848410, 84141805077, 248369601964
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OFFSET
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1,3
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COMMENTS
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Eric Roosendaal counted all admissible sequences up to order j=1000 (2005). Note: there is a typo in both Wagon and Chamberland in the definition of Wagon's constant 9.477955... The expression floor(1+2*i+i*log_2(3)) should be replaced by floor(1+i+i*log_2(3)).
The length of all admissible sequences of order j is A020914(j). - T. D. Noe, Sep 11 2006
Mike Winkler has devised an iterative algorithm that generates the numbers of this sequence using only simple integer arithmetic. See Links, conjecture, proved until n = 107. - Mike Winkler, Jul 23 2011
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REFERENCES
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M. Chamberland, Una actualizacio del problema 3x+1, Butl. Soc. Catalana Mat. 18 (2003) 19-45.
S. Wagon, The Collatz problem, Math. Intelligencer 7 (1985) 72-76.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..500
M. Chamberland, English translation
M. Winkler, On a stopping time algorithm of the 3n+ 1 function.
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FORMULA
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A sequence s(k), where k=1, 2, ..., n, is *admissible* if it satisfies s(k)=3/2 exactly j times, s(k)=1/2 exactly n-j times, s(1)*s(2)*...*s(n) < 1 but s(1)*s(2)*...*s(m) > 1 for all 1 < m < n.
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EXAMPLE
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The unique admissible sequence of order 1 is 3/2, 1/2.
The unique admissible sequence of order 2 is 3/2, 3/2, 1/2, 1/2.
The two admissible sequences of order 3 are 3/2, 3/2, 3/2, 1/2, 1/2 and 3/2, 3/2, 1/2, 3/2, 1/2.
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MATHEMATICA
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(* based on Eric Roosendaal's algorithm *) nn=100; Clear[x, y]; Do[x[i]=0, {i, 0, nn+1}]; x[1]=1; t=Table[Do[y[cnt]=x[cnt]+x[cnt-1], {cnt, p+1}]; Do[x[cnt]=y[cnt], {cnt, p+1}]; admis=0; Do[If[(p+1-cnt)*Log[3]<p*Log[2], admis=admis+x[cnt]; x[cnt]=0], {cnt, p+1}]; admis, {p, 2, nn}]; DeleteCases[t, 0] - T. D. Noe, Sep 11 2006
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CROSSREFS
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Cf. A122790 (Wagon's constant).
Sequence in context: A047749 A134565 * A186009 A034786 A080107 A056156
Adjacent sequences: A100979 A100980 A100981 * A100983 A100984 A100985
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KEYWORD
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nonn
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AUTHOR
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Steven Finch, Jan 13 2005
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EXTENSIONS
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Two more terms from Jules Renucci (jules.renucci(AT)wanadoo.fr), Nov 02 2005
More terms from T. D. Noe, Sep 11 2006
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STATUS
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approved
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