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

2, 3, 5, 6, 7, 11, 12, 13, 19, 20, 21, 29, 30, 31, 41, 42, 43, 55, 56, 57, 71, 72, 73, 89, 90, 91, 109, 110, 111, 131, 132, 133, 155, 156, 157, 181, 182, 183, 209, 210, 211, 239, 240, 241, 271, 272, 273, 305, 306, 307, 341, 342, 343, 379, 380, 381, 419, 420, 421, 461

Offset: 1

Author

Gary E. Davis, Nov 29 2020

Keywords

• nonn
• easy

Comments

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

Examples

			a(1) = 2 since 1st convergent 2, to e, appears at level 2 of the Calkin-Wilf tree.
a(2) = 3 since 2nd convergent 3 appears at level 3, a(3) = 5 since 3rd convergent 8/3 appears at level 5.
		

Links

• Wikipedia, Calkin-Wilf Tree.
• Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Crossrefs

Cf. A002487, A003417 (continued fraction for e), A007676/A007677 (convergents).

Programs

Mathematica

children[{a_,b_}]:={{a,a+b},{a+b,b}};
frac[{a_,b_}]:=a/b;
L[1]={{1,1}};
L[n_]:=Flatten[Map[children,L[n-1]],1];
CWLevel[n_]:=Map[frac,If[n==1,L[1],Complement[L[n],L[n-1]]]];
WhereCW[{a0_,b0_}]:=Module[{a=a0,b=b0,steps},steps =1;While[a>1 || b>1,{a,b}=parent[{a,b}];steps++];steps];
fracpair[k_]:={Numerator[FromContinuedFraction[ContinuedFraction[E,k]]],Denominator[FromContinuedFraction[ContinuedFraction[E,k]]]};
Table[WhereCW[fracpair[k]],{k,1,60}]

PARI

a(n) = sqr(n\3) + n + 1; \\ Kevin Ryde, Dec 26 2020

Formula

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n > 7.

a(n) = floor(n/3)^2 + n + 1. - Kevin Ryde, Dec 26 2020

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