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a(n) is the 2-adic valuation of A173300(n).
2

%I #28 Feb 02 2024 08:28:50

%S 0,0,1,1,2,1,3,3,4,3,5,5,6,5,7,7,8,7,9,9,10,9,11,11,12,11,13,13,14,13,

%T 15,15,16,15,17,17,18,17,19,19,20,19,21,21,22,21,23,23,24,23,25,25,26,

%U 25,27,27,28,27,29,29,30,29,31,31,32,31,33,33,34,33,35,35,36,35,37,37,38,37

%N a(n) is the 2-adic valuation of A173300(n).

%C Conjecture: always follows the pattern A, A, A+1, A, where A is an odd number.

%H Hugo Pfoertner, <a href="/A173989/b173989.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = log(A173300(n))/log(2).

%F Apparently a(n) = A102302(n) for n >= 7. - _Hugo Pfoertner_, Oct 10 2018

%F Conjectures from _Colin Barker_, Oct 10 2018: (Start)

%F G.f.: x^3*(1 + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).

%F a(n) = a(n-1) + a(n-4) - a(n-5) for n > 7.

%F (End)

%p From _R. J. Mathar_, Mar 20 2010: (Start)

%p A173300 := proc(n) local x,y ; x := (1+sqrt(3))/2 ; y := (1-sqrt(3))/2 ; denom(expand(x^n+y^n)) ; end proc:

%p A173989 := proc(n) log[2](A173300(n)) ; end proc: seq(A173989(n),n=3..100) ; (End)

%t Log2[Denominator[Map[First, NestList[{Last[#], Last[#] + First[#]/2} &, {1, 2}, 100]]]] (* Paolo Xausa, Feb 01 2024, after _Nick Hobson_ in A173300 *)

%o (PARI) \\ using _Max Alekseyev_'s function in A173300

%o A173300(n) = denominator(2*polcoeff( lift( Mod((1+x)/2, x^2-3)^n ), 0))

%o for(k=1,74,print1(logint(A173300(k),2),", ")) \\ _Hugo Pfoertner_, Oct 10 2018

%Y Cf. A102302, A173300.

%K nonn

%O 1,5

%A _J. Lowell_, Mar 04 2010

%E More terms from _R. J. Mathar_ and _Max Alekseyev_, Mar 20 2010