A007677 -id:A007677 - cp's OEIS frontend

A120355 a(n) = min{k>0: the n-th convergent to e equals m/k! for some m}.

1, 1, 3, 4, 7, 8, 13, 71, 31, 67, 13, 89, 83, 18089, 5441, 17377, 36269, 26021, 4909, 10391023, 1097, 28879, 1846921, 519691, 1329313, 793279, 7553783, 3308341, 65676881, 662407, 677311, 2425388512913, 4403182913, 10832561

Offset: 0

Jonathan Sondow, Aug 16 2006

			The 6th convergent to e is 87/32 and 32 divides 8! but not 7!, so a(6) = 8.

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

Cf. A002034, A007677.

a(n) = A002034(A007677(n)).

Extended by Max Alekseyev, Jul 28 2009

Missing a(7)=71 inserted by Georg Fischer, Oct 15 2024

A233044 Pairs p, q for those partial sums p/q of the series e = sum_{n>=0} 1/n! that are not convergents to e.

Original entry on oeis.org

1, 1, 5, 2, 65, 24, 163, 60, 1957, 720, 685, 252, 109601, 40320, 98641, 36288, 9864101, 3628800, 13563139, 4989600, 260412269, 95800320, 8463398743, 3113510400, 47395032961, 17435658240, 888656868019, 326918592000

Offset: 1

Jonathan Sondow, Dec 07 2013

nonn

Sondow (2006) conjectured that 2/1 and 8/3 are the only partial sums of the Taylor series for e that are also convergents to the simple continued fraction for e. Sondow and Schalm (2008, 2010) proved partial results toward the conjecture. Berndt, Kim, and Zaharescu (2012) proved it in full.

			1/1, 5/2, 65/24, 163/60, 1957/720, 685/252, 109601/40320, 98641/36288, 9864101/3628800, 13563139/4989600, 260412269/95800320, 8463398743/3113510400, 47395032961/17435658240, 888656868019/326918592000

J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e?, (and a link to the primes 2, 5, 13, 37, 463), part I, in Tapas in Experimental Mathematics, T. Amdeberhan and V. H. Moll, eds., Contemp. Math., vol. 457, American Mathematical Society, Providence, RI, 2008, pp. 273-284.

B. Berndt, S. Kim, and A. Zaharescu, Diophantine approximation of the exponential function and Sondow's conjecture, abstract 2012.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly, 113 (2006), 637-641.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e?, (and a link to the primes 2, 5, 13, 37, 463), part II, in Gems in Experimental Mathematics, T. Amdeberhan, L. A. Medina, V. H. Moll, eds., Contemp. Math., vol. 517, American Mathematical Society, Providence, RI, 2010, pp. 349-363.

Cf. A061354, A061355, A007676, A007677.

a(2n-1)/a(2n) = A061354(k)/A061355(k) for some k <> 1 and 3.

a(2n-1)/a(2n) <> A007676(k)/A007677(k) for all k.

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

Original entry on oeis.org

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

Gary E. Davis, Nov 29 2020

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

			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.

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

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

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}]

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

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

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A120355 a(n) = min{k>0: the n-th convergent to e equals m/k! for some m}.

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A233044 Pairs p, q for those partial sums p/q of the series e = sum_{n>=0} 1/n! that are not convergents to e.

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

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