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3, 9, 33, 129, 513, 2049, 8193, 32769, 131073, 524289, 2097153, 8388609, 33554433, 134217729, 536870913, 2147483649, 8589934593, 34359738369, 137438953473, 549755813889, 2199023255553
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Number of pairs of polynomials (f,g) in GF(2)[x] satisfying deg(f) <=n, deg(g) <= n and gcd(f,g) = 1.
An unpublished result due to Stephen Suen, David desJardin and W. Edwin Clark. This the case k = 2, q = 2 of their formula q^((n+1)*k) * (1 - 1/q^(k-1) + (q-1)/q^((n+1)*k)) for the number of ordered k-tuples (f_1, ..., f_k) of polynomials in GF(q)[x] such that deg(f_i) <= n for all i and gcd((f_1, ..., f_k) = 1
Apparently the same as A084508 shifted left.
Terms in binary are palindromes of the form 1x1 where x is a string of 2*n zeros (A152577). - Brad Clardy, Sep 01 2011
For n > 0, a(n) is the number k such that the number of iterations of the map k -> (3k +1)/8 == 4 (mod 8) until reaching (3k +1)/8 <> 4 (mod 8) equals n. (see the Collatz problem : the start of the parity trajectory of a(n) is n times {100} = 100100100100…100abcd…).- Michel Lagneau, Jan 23 2012
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index to sequences with linear recurrences with constant coefficients, signature (5,-4).
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FORMULA
| G.f.: (3-6*x)/((1-x)*(1-4*x)).
a(n) = 3 *A007583(n).
a(n) = 4*a(n-1) - 3. - Lekraj Beedassy , Apr 29 2005
a(n) = A099393(n+1) - 2*A099393(n). - Brad Clardy, Sep 01 2011
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EXAMPLE
| a(0) = 3 since there are three pairs, (0,1), (1,0) and (1,1) of polynomials (f,g) in GF(2)[x] of degree at most 0 such that gcd(f,g) = 1.
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PROG
| (MAGMA) [2^(2*n+1) + 1: n in [0..30]]; // Vincenzo Librandi, May 16 2011
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CROSSREFS
| Cf. A087290, A087291, A087292, A099393.
Equals A004171 + 1.
Sequence in context: A151040 A151041 A151042 * A084508 A151043 A151044
Adjacent sequences: A087286 A087287 A087288 * A087290 A087291 A087292
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KEYWORD
| easy,nonn
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AUTHOR
| W. Edwin Clark (eclark(AT)math.usf.edu), Aug 29 2003
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