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A339267 Level of the Calkin-Wilf tree in which the n-th convergent of the continued fraction for e appears.

%I A339267 #24 Dec 27 2020 19:31:14
%S A339267 2,3,5,6,7,11,12,13,19,20,21,29,30,31,41,42,43,55,56,57,71,72,73,89,
%T A339267 90,91,109,110,111,131,132,133,155,156,157,181,182,183,209,210,211,
%U A339267 239,240,241,271,272,273,305,306,307,341,342,343,379,380,381,419,420,421,461
%N A339267 Level of the Calkin-Wilf tree in which the n-th convergent of the continued fraction for e appears.
%C A339267 The depth level of a rational in the Calkin-Wilf tree is the sum of its continued fraction terms, with the root (1/1) as level 1 for this purpose.  So the present sequence is partial sums of the continued fraction terms of e (A003417).  Depth levels are the same in the related trees Stern-Brocot, Bird, etc. - _Kevin Ryde_, Dec 26 2020
%H A339267 Wikipedia, <a href="https://en.wikipedia.org/wiki/Calkin%E2%80%93Wilf_tree">Calkin-Wilf Tree</a>.
%H A339267 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F A339267 a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n > 7.
%F A339267 a(n) = floor(n/3)^2 + n + 1. - _Kevin Ryde_, Dec 26 2020
%F A339267 G.f.: x*(2 + x + 2*x^2 - 3*x^3 - x^4 + x^6)/((1 - x)^3*(1 + x + x^2)^2). - _Stefano Spezia_, Dec 27 2020
%e A339267 a(1) = 2 since 1st convergent 2, to e, appears at level 2 of the Calkin-Wilf tree.
%e A339267 a(2) = 3 since 2nd convergent 3 appears at level 3, a(3) = 5 since 3rd convergent 8/3 appears at level 5.
%t A339267 children[{a_,b_}]:={{a,a+b},{a+b,b}};
%t A339267 frac[{a_,b_}]:=a/b;
%t A339267 L[1]={{1,1}};
%t A339267 L[n_]:=Flatten[Map[children,L[n-1]],1];
%t A339267 CWLevel[n_]:=Map[frac,If[n==1,L[1],Complement[L[n],L[n-1]]]];
%t A339267 WhereCW[{a0_,b0_}]:=Module[{a=a0,b=b0,steps},steps =1;While[a>1 || b>1,{a,b}=parent[{a,b}];steps++];steps];
%t A339267 fracpair[k_]:={Numerator[FromContinuedFraction[ContinuedFraction[E,k]]],Denominator[FromContinuedFraction[ContinuedFraction[E,k]]]};
%t A339267 Table[WhereCW[fracpair[k]],{k,1,60}]
%o A339267 (PARI) a(n) = sqr(n\3) + n + 1; \\ _Kevin Ryde_, Dec 26 2020
%Y A339267 Cf. A002487, A003417 (continued fraction for e), A007676/A007677 (convergents).
%K A339267 nonn,easy
%O A339267 1,1
%A A339267 _Gary E. Davis_, Nov 29 2020