A113013 -id:A113013 - cp's OEIS frontend

A113011 Decimal expansion of 1/(sqrt(e) - 1).

1, 5, 4, 1, 4, 9, 4, 0, 8, 2, 5, 3, 6, 7, 9, 8, 2, 8, 4, 1, 3, 1, 1, 0, 3, 4, 4, 4, 4, 7, 2, 5, 1, 4, 6, 3, 8, 3, 4, 0, 4, 5, 9, 2, 3, 6, 8, 4, 1, 8, 8, 2, 1, 0, 9, 4, 7, 4, 1, 3, 6, 9, 5, 6, 6, 3, 7, 5, 4, 2, 6, 3, 9, 1, 4, 3, 3, 1, 4, 8, 0, 7, 0, 7, 1, 8, 2, 5, 7, 2, 4, 0, 8, 5, 0, 0, 7, 7, 4, 2, 2, 4

Offset: 1

Eric W. Weisstein, following a suggestion of Grover W. Hughes, Oct 09 2005

Has continued fraction 1+2/(3+4/(5+6/7+...)).

Simple continued fraction is 1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, {1, 4k+1, 1}, ..., . - Robert G. Wilson v, Jul 01 2007

			1.54149408253679828413110344447251463834045923684188210947413695663...

G. C. Greubel, Table of n, a(n) for n = 1..20000
Itay Beit-Halachmi and Ido Kaminer, The Ramanujan Library -- Automated Discovery on the Hypergraph of Integer Relations, arXiv:2412.12361 [cs.AI], 2024. See p. 6.
Leonhard Euler, On the formation of continued fractions, arXiv:math/0508227 [math.HO], 2005, see p. 14.
Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18 - 29 mai 2015: Oujda (Maroc).
Eric Weisstein's World of Mathematics, Continued Fraction.
Index entries for transcendental numbers.

Cf. A019774, A113012, A113013.

1/(Sqrt(Exp(1)) - 1); // G. C. Greubel, Apr 09 2018

First@ RealDigits[ 1 / (Exp[1/2] - 1), 10, 111] (* Robert G. Wilson v, Jul 01 2007 *)
f[n_] := Fold[ Last@ #2 + First@ #2/#1 &, 2n - 1, Partition[ Reverse@ Range[ 2n - 2], 2]]; RealDigits[ f[61], 10, 105][[1]] (* Robert G. Wilson v, Jul 07 2012 *)
Rest[realDigitsRecip[Sqrt[E]-1]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Nov 04 2024 *)

1/(sqrt(exp(1)) - 1) \\ G. C. Greubel, Apr 09 2018

Equals Integral_{x = 0..oo} floor(2*x)*exp(-x) dx. - Peter Bala, Oct 09 2013
Equals 3/2 + Sum_{k>=0} tanh(1/2^(k+3))/2^(k+2). - Antonio Graciá Llorente, Jan 21 2024
Conjecture: 1/(sqrt(e) - 1) = 1 + K_{n>=1} 2*n/(4*n^2-1)/1, where K is the Gauss notation for an infinite continued fraction. In the expanded form, 1/(sqrt(e) - 1) = 1 + 2/3/(1 + 4/15/(1 + 6/35/(1 + ...))) (see Beit-Halachmi and Kaminer). - Stefano Spezia, Dec 27 2024
Equals 1/(A019774 - 1). - Hugo Pfoertner, Dec 27 2024

Simpler definition from T. D. Noe, Oct 09 2005

Euler reference from David L. Harden, Oct 09 2005

A113012 Numerators of convergents to 1 + 2/(3 + 4/(5 + 6/(7 + ...))).

Original entry on oeis.org

1, 5, 29, 233, 2329, 27949, 78257, 6260561, 112690097, 2253801941, 49583642701, 47600296993, 30940193045449, 866325405272573, 25989762158177189, 831672389061670049, 5655372245619356333, 1017967004211484139941, 38682746160036397317757, 1547309846401455892710281

Offset: 0

Eric W. Weisstein, Oct 10 2005

			1, 5/3, 29/19, 233/151, 2329/1511, ...

G. C. Greubel, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Cf. A000354, A113011, A113013.

List(List([0..20],n->Sum([0..n],k->(-1)^k*(1/(Product([0..Int(2*k/2)-1],i->2*k-2*i))))),NumeratorRat); # Muniru A Asiru, Apr 14 2018

Numerator[Table[Sum[(-1)^k*1/(2k)!!,{k,0,n}],{n,1,25}]] (* Alexander Adamchuk, Jul 02 2006 *)
f[n_] := Fold[ Last@ #2 + First@ #2/#1 &, 2n - 1, Partition[ Reverse@ Range[ 2n - 2], 2]]; Numerator[ Array[ f, 18]]  (* Robert G. Wilson v, Jul 07 2012 *)
a[ n_] := If[ n < 0, 0, Numerator[ 1 + ContinuedFractionK[2 i, 2 i + 1, {i, 1, n}]]]; (* Michael Somos, Apr 14 2018 *)
Table[1 + ContinuedFractionK[2 k, 2 k + 1, {k, n}], {n, 0, 20}] // Numerator (* Eric W. Weisstein, Apr 14 2018 *)
Table[1/((Sqrt[E] Gamma[n + 2])/Gamma[n + 2, -1/2] - 1), {n, 0, 20}] // Numerator (* Eric W. Weisstein, Apr 14 2018 *)

{a(n) = my(A); if( n<0, 0, A = contfracpnqn( matrix(2, n, j, i, [2*i, 2*i+1] [j]) ); numerator( 1 + A[2, 1] / A[1, 1]) )}; /* Michael Somos, Apr 14 2018 */

a(n) = Numerator(Sum_{k=0..n+1} (-1)^k*1/(2k)!!). - Alexander Adamchuk, Jul 02 2006

a(n) <= A000354(n+1). - Michael Somos, Sep 28 2017

A354299 a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

Original entry on oeis.org

1, 3, 15, 105, 189, 10395, 135135, 2027025, 34459425, 130945815, 13749310575, 316234143225, 7905853580625, 12556355686875, 1238056670725875, 776918153694375, 6332659870762850625, 7642865361265509375, 8200794532637891559375, 63966197354575554163125, 13113070457687988603440625

Offset: 1

Ilya Gutkovskiy, May 23 2022

			1, 2/3, 11/15, 76/105, 137/189, 7534/10395, 97943/135135, 1469144/2027025, 24975449/34459425, ...

Robert Israel, Table of n, a(n) for n = 1..403

Cf. A001147, A053556, A061355, A064647, A113013, A143383, A289488, A306858, A354298 (numerators).

S:= 0: R:= NULL:
for n from 1 to 100 do
  S:= S + (-1)^(n+1)/doublefactorial(2*n-1);
  R:= R, denom(S);
od:
R; # Robert Israel, Jan 10 2024

Table[Sum[(-1)^(k + 1)/(2 k - 1)!!, {k, 1, n}], {n, 1, 21}] // Denominator
nmax = 21; CoefficientList[Series[Sqrt[Pi x Exp[-x]/2] Erfi[Sqrt[x/2]]/(1 - x), {x, 0, nmax}], x] // Denominator // Rest
Table[1/(1 + ContinuedFractionK[2 k - 1, 2 k, {k, 1, n - 1}]), {n, 1, 21}] // Denominator

Denominators of coefficients in expansion of sqrt(Pi*x*exp(-x)/2) * erfi(sqrt(x/2)) / (1 - x).

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A113011 Decimal expansion of 1/(sqrt(e) - 1).

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A113012 Numerators of convergents to 1 + 2/(3 + 4/(5 + 6/(7 + ...))).

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A354299 a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

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