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A131450 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 4
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; text; internal format)
OFFSET

1,9

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.

With a(2) = 0 this is the first difference sequence of A060322, the row length sequence of A248573 (Collatz-Terras tree). - Wolfdieter Lang, May 04 2015

LINKS

Table of n, a(n) for n=1..50.

Perry Dobbie, Collatz representations.

Index entries for sequences related to 3x+1 (or Collatz) problem

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.

CROSSREFS

Cf. A060322, A248573.

Sequence in context: A078611 A211376 A278249 * A114218 A133691 A111973

Adjacent sequences:  A131447 A131448 A131449 * A131451 A131452 A131453

KEYWORD

nonn

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

EXTENSIONS

Edited by David Applegate, Oct 16 2008

STATUS

approved

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Last modified February 24 22:57 EST 2020. Contains 332216 sequences. (Running on oeis4.)