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A131450
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a(n) = number of integers x that can be written x=(2^c[1] - 2^c[2] -3*2^c[3] - 3^2*2^c[4] - ... - 3^(m-2)*2^c[m] - 3^(m-1) ) / 3^m for integers c[1], c[2], ..., c[m] such that n=c[1]>c[2]>...>c[m]>0 and c[1] - c[2] != 2 if m >= 2
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3
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0, 1, 0, 1, 1, 1, 1, 1, 2, 4, 6, 6, 7, 8, 11, 18, 23, 29, 39, 52, 71, 99, 124, 160, 220, 302, 403, 532, 707, 936, 1249, 1668, 2220, 2976, 3966, 5278, 7028, 9386, 12531, 16696, 22246, 29622, 39540, 52768, 70395, 93795, 124977, 166619, 222222, 296358
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,9
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COMMENTS
| For m = 1, the expression for x becomes x=(2^c[1] - 1) / 3.
Also the number of odd x with stopping time n for the Collatz or 3x+1 problem where x->x/2 if x is even, x->(3x+1)/2 if x is odd (see A060322), except that 1 is counted as having stopping time 2 instead of 0.
Equivalently, a(n) is the number of x == 2 (mod 3) with stopping time n-1.
The number of possible c[1],...,c[m] is 2^(n-1)-2^(n-3); most do not yield integer x.
n-c[m], n-c[m-1], ..., n-c[2] are the stopping times of the odd integers in the Collatz trajectory of x.
a(n) = a(n-2) + a(n-2):(x is 1 mod 6) + a(n-1):(x is 5 mod 6)
It is conjectured that a(n)/a(n-1) -> 4/3 as n-> infinity.
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LINKS
| Perry Dobbie, Collatz representations.
Index entries for sequences related to 3x+1 (or Collatz) problem
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EXAMPLE
| For n=3, the only valid c are:
c=(3,2,1) (2^3 - 2^2 - 3^1*2^1 - 3^2) / 3^3 = -11/27,
c=(3,2) (2^3 - 2^2 - 3^1) / 3^2 = 1/9,
c=(3) (2^3 - 2^0 ) / 3 = 7/3,
and none are integers so a(3) = 0.
a(9)=2
c=(9,5) (2^9 - 2^5 - 3) / 3 = 53
c=(9,5,2) (2^9 - 2^5 - 3*2^2 - 9) / 27 = 17
and no other valid c give integer x.
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CROSSREFS
| Sequence in context: A111150 A166983 A078611 * A114218 A111973 A133691
Adjacent sequences: A131447 A131448 A131449 * A131451 A131452 A131453
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KEYWORD
| nonn
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AUTHOR
| Perry Dobbie (pdobbie(AT)rogers.com), Jul 11 2007, Jul 12 2007, Jul 13 2007, Jul 17 2007, Jul 22 2007, Oct 15 2008
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EXTENSIONS
| Edited by David Applegate, Oct 16 2008
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