A007676 -id:A007676 - cp's OEIS frontend

A085369 Cutting sequence for 1/e.

1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0

Offset: 1

Gary W. Adamson, Jun 26 2003

nonn

Through any A085368(n) number of terms in the cutting sequence, A007677(n-1) of those terms are zeros and A007676(n) are ones. Check: A085368(5) = 26, the sequence being 3, 4, 11, 15, 26, ... (sum of numerators and denominators of convergents to 1/e). Then through n=26, A085369(n) is 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0, with 7 zeros and 19 ones, (7/19 being the 5th convergent to 1/e): 7/19 = [2, 1, 2, 1, 1]. Numerator and denominator sum = 26, with 7 zeros and 19 ones, with the zeros occupying positions n = 3, 7, 11, 14, 18, 22 and 26 (also being the first 7 terms of A000572). Positions of the cutting sequence occupied by ones (1, 2, 4, 5, 6, ...) are consecutive terms of the lower Beatty sequence A006594, being generated by floor(n*(1 + 1/e)).

			a(6) = 1, where 1's correspond to members of the lower Beatty pair A006594 which is generated from floor(n*(1 + 1/e)). Check: floor(5*(1 + 1/e)) = 6. All terms not in A006594 are 0's.
a(7) = 0, where 7 is not a member of A006594, but is a member of the upper Beatty pair sequence A000572 which has the generator floor(n*(e + 1)). Check: floor(2*(1 + e)) = 7.

Manfred R. Schroeder, "Fractals, Chaos, Power Laws", Freeman, 1996, p. 56.

Cf. A085368, A007677, A007676, A006594, A000572.

Given the line y = (1/e)x starting from (0, 0) and passing through an array of squares, a "1" denotes an intersection with a vertical line, while an "0" denotes an intersection with a horizontal line.

n for 0's are consecutive terms of upper Beatty pair terms A000572: 3, 7, 11, 14, 18, 22, 26, ..., while n's for all 1's are paired lower Beatty terms of A006594: 1, 2, 4, 5, 6, 8, ...

A259589 Numerators of the other-side convergents to e.

Original entry on oeis.org

3, 5, 11, 19, 30, 106, 193, 299, 1457, 2721, 4178, 25946, 49171, 75117, 566827, 1084483, 1651310, 14665106, 28245729, 42910835, 438351041, 848456353, 1286807394, 14862109042, 28875761731, 43737870773, 563501581931, 1098127402131, 1661628984062

Offset: 0

Clark Kimberling, Jul 17 2015

Suppose that a positive irrational number r has continued fraction [a(0), a(1), ... ]. Define sequences p(i), q(i), P(i), Q(i) from the numerators and denominators of finite continued fractions as follows:
p(i)/q(i) = [a(0), a(1), ... a(i)] and P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1]. The fractions p(i)/q(i) are the convergents to r, and the fractions P(i)/Q(i) are introduced here as the "other-side convergents" to
r, because p(2k)/q(2k) < r < P(2k)/Q(2k) and P(2k+1)/Q(2k+1) < r < p(2k+1)/q(2k+1), for k >= 0.
Closeness of P(i)/Q(i) to r is indicated by |r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.

			For r = e, the first 13 other-side convergents are 3/1, 5/2, 11/4, 19/7, 30/11, 106/39, 193/71, 299/110, 1457/536, 2721/1001, 4178/1537, 25946/9545, 49171/18089.
A comparison of convergents with other-side convergents:
i  p(i)/q(i)       P(i)/Q(i)  p(i)*Q(i) - P(i)*q(i)
0     2/1    < e <    3/1               -1
1     3/1    > e >    5/2                1
2     8/3    < e <   11/4               -1
3    11/4    > e >   19/7                1
4    19/7    < e <   30/11              -1
5    87/32   > e >  106/39               1

Cf. A259588, A007676, A007677.

r = E; a[i_] := Take[ContinuedFraction[r, 35], i];
b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
u = Denominator[t]  (* A259588 *)
v = Numerator[t]    (* A259589 *)

p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i).

A078976 Numerator of n-th convergent to e^(2/3).

Original entry on oeis.org

1, 2, 37, 261, 298, 559, 5888, 318511, 5102064, 5420575, 10522639, 205350716, 18492087079, 462507527691, 480999614770, 943507142461, 26899199603678, 3390242657205889, 115295149544603904, 118685392201809793, 233980541746413697, 8775965436819116582, 1421940381306443299981

Offset: 1

Benoit Cloitre, Dec 19 2002

Cf. A069951, A001518, A007676, A078977 (denominators).

Convergents[Exp[2/3], 25] // Numerator (* Amiram Eldar, May 09 2025 *)

default(realprecision,100); /* large enough */
a(n)=contfracpnqn(contfrac(exp(2/3), n))[1,1]
vector(30,n,a(n))

Special cases : a(5k+1) = A001518(3k); a(5k+3) = A001518(3k+2).

A367328 The x-coordinate of the point where x + y = n, x and y are integers and x/y is as close as possible to e.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 34, 34, 35, 36, 37, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 48, 48, 49, 50

Offset: 1

Colin Linzer, Nov 14 2023

a(n) is nondecreasing; lim_{n->oo} a(n) = oo.

Swapping the x and y coordinate of the sequence does not yield the sequence defined as the point where x + y = n, x and y are integers and x/y is as close as possible to 1/e even when excluding terms that would lead to a division by 0.

			For n = 3, the possible points are (0,3), (1,2), (2,1) as any negative value would would be further from e than 0/3. The closest value to e out of these is 2/1 so a(3) = 2.

Cf. A001113 (e), A367329 (y-coordinate), A007676.

a(n) is always either ceiling(n*e/(1 + e)) or floor(n*e/(1 + e)) = A076538(n).

A114539 Number of correct decimal digits given by the n-th convergent to e.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 8, 8, 9, 10, 10, 12, 13, 13, 15, 16, 16, 18, 19, 19, 21, 22, 22, 24, 26, 26, 27, 29, 29, 31, 32, 32, 34, 36, 36, 38, 39, 39, 41, 43, 43, 45, 47, 47, 48, 50, 50, 52, 54, 54, 56, 58, 58, 60, 62, 62, 64, 65, 66, 67, 69, 69, 71, 73, 73, 75, 77, 77

Offset: 0

Eric W. Weisstein, Dec 07 2005

Eric Weisstein's World of Mathematics, e Continued Fraction

Cf. A007676, A007677.

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

cp's OEIS Frontend

A085369 Cutting sequence for 1/e.

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A259589 Numerators of the other-side convergents to e.

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A078976 Numerator of n-th convergent to e^(2/3).

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A367328 The x-coordinate of the point where x + y = n, x and y are integers and x/y is as close as possible to e.

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A114539 Number of correct decimal digits given by the n-th convergent to e.

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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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