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A341313 a(n) = (a(n-1) + a(n-3))/2^m, where 2^m is the highest power of 2 that divides both a(n-1) and a(n-3), with a(0) = a(1) = a(2) = 1.

%I A341313 #13 Feb 17 2021 20:25:04
%S A341313 1,1,1,2,3,4,3,6,5,8,7,12,5,12,6,11,23,29,40,63,92,33,96,47,80,11,58,
%T A341313 69,80,69,138,109,178,158,267,445,603,870,1315,1918,1394,2709,4627,
%U A341313 6021,8730,13357,19378,14054,27411,46789,60843,88254,135043,195886,142070,277113,472999
%N A341313 a(n) = (a(n-1) + a(n-3))/2^m, where 2^m is the highest power of 2 that divides both a(n-1) and a(n-3), with a(0) = a(1) = a(2) = 1.
%C A341313 A sequence intermediate between Narayana's A000930 and Reed Kelly's A214551.
%C A341313 It will be interesting to compare the growth rates of A000930 (well-understood), A241551 (a mystery), the present sequence, and A341312.
%C A341313 It appears that the equation log(a(n)) = 0.265986*n + 1.56445 is a good fit to the data (see the figures). - _Hugo Pfoertner_, Feb 17 2021
%H A341313 Hugo Pfoertner, <a href="/A341313/b341313.txt">Table of n, a(n) for n = 0..5000</a>
%H A341313 Hugo Pfoertner, <a href="/A341312/a341312.pdf">Comparison of linear fits to logarithm of A341312, A341313, A214551 (Reed Kelly), and A000930 (Narayana's cows)</a>.
%H A341313 Hugo Pfoertner, <a href="/A341313/a341313.pdf">Deviation of log(A341313) from linear fit in range 3...10000</a>.
%p A341313 RK3:=proc(n) local t1,t2; option remember;
%p A341313 if n <= 2 then 1 else t1:=RK3(n-3)+RK3(n-1);
%p A341313 t2 := min( padic[ordp](RK3(n-3), 2), padic[ordp](RK3(n-1), 2) );
%p A341313 t1/2^t2;
%p A341313 fi;
%p A341313 end;
%p A341313 [seq(RK3(n),n=0..60)];
%o A341313 (PARI) a341313(nterms)={my(a=vector(nterms));a[1]=a[2]=1;a[3]=2;for(n=4,nterms,a[n]=(a[n-1]+a[n-3])/2^min(valuation(a[n-1],2),valuation(a[n-3],2)));concat([1],a)};
%o A341313 a341313(60) \\ _Hugo Pfoertner_, Feb 16 2021
%Y A341313 Cf. A000930, A214551, A341312.
%K A341313 nonn
%O A341313 0,4
%A A341313 _N. J. A. Sloane_, Feb 16 2021