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%I #86 Aug 20 2022 13:30:42
%S 3,9,33,129,513,2049,8193,32769,131073,524289,2097153,8388609,
%T 33554433,134217729,536870913,2147483649,8589934593,34359738369,
%U 137438953473,549755813889,2199023255553
%N a(n) = 2^(2*n+1) + 1.
%C Number of pairs of polynomials (f,g) in GF(2)[x] satisfying deg(f) <= n, deg(g) <= n and gcd(f,g) = 1.
%C An unpublished result due to Stephen Suen, _David desJardins_, and _W. Edwin Clark_. This is 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.
%C Apparently the same as A084508 shifted left.
%C Terms in binary are palindromes of the form 1x1 where x is a string of 2*n zeros (A152577). - _Brad Clardy_, Sep 01 2011
%C 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
%C An Engel expansion of 2 to the base 4 as defined in A181565, with the associated series expansion 2 = 4/3 + 4^2/(3*9) + 4^3/(3*9*33) + 4^4/(3*9*33*129) + .... Cf. A199561 and A207262. - _Peter Bala_, Oct 29 2013
%C For x = A083420(n), y = A000079(n+1), z = a(n) then x^2 + 2*y^2 = z^2. - _Vincenzo Librandi_, Jun 09 2014
%C A254046(n+1) is the 3-adic valuation of a(n). - _Fred Daniel Kline_, Jan 11 2017
%H Vincenzo Librandi, <a href="/A087289/b087289.txt">Table of n, a(n) for n = 0..1000</a>
%H K. Morrison, <a href="https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.701.4944&rep=rep1&type=pdf">Random polynomials over finite fields</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).
%F G.f.: (3-6*x)/((1-x)*(1-4*x)).
%F a(n) = 3 *A007583(n).
%F a(n) = 4*a(n-1) - 3. - _Lekraj Beedassy_, Apr 29 2005
%F a(n) = A099393(n+1) - 2*A099393(n). - _Brad Clardy_, Sep 01 2011
%F a(n) = 2^(2*n + 1) * a(-1-n) for all n in Z. - _Michael Somos_, Jan 11 2017
%F a(n) = A283070(n) - 1. - _Peter M. Chema_, Mar 02 2017
%e 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.
%t Table[2^(2 n + 1) + 1, {n, 0, 20}] (* or *) 3 NestList[4 # - 1 &, 1, 20]
%t (* or *) CoefficientList[Series[(3 - 6 x)/((1 - x) (1 - 4 x)), {x, 0, 20}], x] (* _Michael De Vlieger_, Mar 03 2017 *)
%o (Magma) [2^(2*n+1) + 1: n in [0..30]]; // _Vincenzo Librandi_, May 16 2011
%o (PARI) a(n)=2^(2*n+1)+1 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A087290, A087291, A087292, A099393.
%Y Equals A004171 + 1.
%Y Cf. also A181565, A199561, A207262, A007583, A283070, A254046.
%K easy,nonn
%O 0,1
%A _W. Edwin Clark_, Aug 29 2003
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