A113011 -id:A113011 - cp's OEIS frontend

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

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

A113013 Denominators of convergents to 1 + 2/(3 + 4/(5 + 6/(7 + ...))).

Original entry on oeis.org

1, 3, 19, 151, 1511, 18131, 50767, 4061359, 73104463, 1462089259, 32165963699, 30879325151, 20071561348151, 562003717748227, 16860111532446811, 539523569038297951, 3668760269460426067, 660376848502876692059, 25094320243109314298243, 1003772809724372571929719, 42158458008423648021048199

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. A113011, A113012.

f[n_] := Fold[ Last@ #2 + First@ #2/#1 &, 2n - 1, Partition[ Reverse@ Range[ 2n - 2], 2]]; Denominator[ Array[ f, 18]] (* Robert G. Wilson v, Jul 07 2012 *)
(* It is interesting to note that FoldList[2 #1*#2 - (-1)^#2 &, 0, Range[19]] matches many of the terms. - Robert G. Wilson v, Jul 07 2012 *)
a[ n_] := If[ n < 0, 0, Denominator[ 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}] // Denominator (* Eric W. Weisstein, Apr 14 2018 *)
Table[1/((Sqrt[E] Gamma[n + 2])/Gamma[n + 2, -1/2] - 1), {n, 0, 20}] // Denominator (* 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]) ); denominator( 1 + A[2, 1] / A[1, 1]) )}; /* Michael Somos, Apr 14 2018 */

a(n) = denominators of 1/((sqrt(e) * Gamma(n+2))/Gamma(n+2, -1/2) - 1), where Gamma(x, a) is the incomplete Gamma function. - Eric W. Weisstein, Apr 14 2018

A365307 Decimal expansion of 1/(2*e-5).

Original entry on oeis.org

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

Offset: 1

Rok Cestnik, Aug 31 2023

The continued fraction expansion is A081750 with initial term 5 omitted.

			2.2906166927853624221...

Cf. A001113, A194807, A019762, A113011, A081750, A073333, A185108.

A365307 = RealDigits[N[1/(2*E-5),#+1]][[1]][[1;;-2]]&;

PARI
```
1/(2*exp(1)-5).
```

Equals 2 + 1/(3 + 2/(4 + 3/(5 + 4/(6 + 5/( ... /(n+1 + n/(n+2 + ... ))))))).
From Peter Bala, Oct 23 2023: (Start)
Define s(n) = Sum_{k = 3..n} 1/k!. Then 1/(2*e - 5) = 3 - (1/2)*Sum_{n >= 3 } 1/( (n+1)!*s(n)*s(n+1) ) is a rapidly converging series of rationals. Cf. A073333 and A194807.
Equivalently, 1/(2*e - 5) = 3 - (1/2)*(3!/(1*5) + 4!/(5*26) + 5!/(26*157) + 6!/(157*1100) + ...), where [1, 5, 26, 157, 1100, ... ] is A185108. (End)

A367120 Decimal expansion of continued fraction 2+1/(4+3/(6+5/(8+7/(...)))).

Original entry on oeis.org

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

Offset: 1

Rok Cestnik, Nov 13 2023

			2.224412437956340467163837541384021939...

Cf. A113014, A113011, A194807, A365307.

First[RealDigits[2/HypergeometricPFQ[{1, 1}, {3/2, 3}, -1/2], 10, 100]] (* or *)
First[RealDigits[2/Sum[(-1)^k/Binomial[k+2, 2]/(2*k+1)!!, {k, 0, Infinity}], 10, 100]] (* Paolo Xausa, Nov 18 2024 *)

N=50;
doblfac(n) = if(n<0, 0, n<2, 1, n*doblfac(n-2));
ap1 = 2 / sum(k=0,N, (-1)^k/binomial(k+2,2)/doblfac(2*k+1));
ap2 = 2 / sum(k=0,N+1, (-1)^k/binomial(k+2,2)/doblfac(2*k+1));
n = 0; while(digits(floor(10^(n+1)*ap1)) == digits(floor(10^(n+1)*ap2)), n++);
A367120 = digits(floor(10^n*ap1));

Equals 2 / pFq(1,1; 3/2,3; -1/2) where pFq() is the generalized hypergeometric function.

Equals 2 / Sum_{k>=0} (-1)^k/binomial(k+2,2)/(2*k+1)!! = 2 / (1 - 1/9 + 1/90 - 1/1050 + 1/14175 - 1/218295 + ... ).

A130701 Decimal expansion of mu, a continued fraction first constructed from the Fibonacci numbers (A000045).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 01 2007

1+ 1/(2 +3/(5 + 8/(13 + ... .
Cited inaccurately in the second reference because they used only the first 28 terms.
Mu's simple continued fraction: 1, 2, 1, 1, 6, 3, 1, 1, 3, 2, 2, 2, 9, 1, 1, 1, 13, 1, 1, 189, 1, 10, 2, 6, 1, 6, 1, 5, 1, 59, 4, 24, 1, 42, 2, 1, 59, 1, 1, 2, 1, ....
Mu's increasingly larger PQ's: 1, 2, 6, 9, 13, 189, 1138, 4150, 6165, 90642, 90676, 526142, 757765, 20411415, 35535156, 271384175, ..., at positions: 1, 2, 5, 13, 17, 20, 454, 529, 708, 1832, 9248, 9631, 211052, 552035, 4552470, 4928425, (9290954)....

Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, Prometheus Books, NY, 2007, page 171.

Joseph S. Madachy, A Fibonacci constant, Recreational Mathematics, Fibonacci Quarterly 6:6 (1968), p. 385.

Cf. A000045, A113011.

First@ RealDigits@ N[ Fold[ Last@#2 + First@#2/#1 &, 1, Partition[ Reverse@ Fibonacci@ Range@48, 2]], 111]

A131701 Decimal expansion of the continued fraction 4+6/(9+10/(14+15/21+...)) where the terms are the semiprimes: A001358.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post & Robert G. Wilson v, Sep 08 2007

			4.6197768903303689380682119968437423840172825564412923389842082329...

Cf. A001358, A113011.

semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; lst = Select[Range@ 400, semiPrimeQ@ # &]; First@ RealDigits@ N[ Fold[ Last@ #2 + First@ #2/#1 &, 4, Partition[Reverse@ lst, 2]], 111] (* Robert G. Wilson v *)

A181050 Decimal expansion of the constant 1+3/(5+7/(9+11/(13+...))), using all odd integers in this generalized continued fraction.

Original entry on oeis.org

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

Offset: 1

Jonathan D. B. Hodgson, Oct 01 2010

The (simple) continued fraction of this constant is [1;1,1,9,1,1,17,1,1,25,...], every 3rd term being of the form 8n+1.

			1.524965344417894912821223094...

Cf. A113011.

r:= (n, i)-> n+ `if`(i<1, 1, (n+2)/r(n+4, i-1)):
s:= convert(evalf(r(1, 80)/10, 130), string):
seq(parse(s[n+1]), n=1..120); # Alois P. Heinz, Oct 16 2011

digits = 105; f[n_] := f[n] = Fold[#2 + (#2+2)/#1 &, 4*n+1, Range[4*n-3, 1, -4] ] // RealDigits[#, 10, digits]& // First; f[digits]; f[n = 2*digits]; While[f[n] != f[n/2], n = 2*n]; f[n] (* Jean-François Alcover, Feb 21 2014 *)

cp's OEIS Frontend

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

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A113013 Denominators of convergents to 1 + 2/(3 + 4/(5 + 6/(7 + ...))).

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A365307 Decimal expansion of 1/(2*e-5).

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A367120 Decimal expansion of continued fraction 2+1/(4+3/(6+5/(8+7/(...)))).

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A130701 Decimal expansion of mu, a continued fraction first constructed from the Fibonacci numbers (A000045).

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A131701 Decimal expansion of the continued fraction 4+6/(9+10/(14+15/21+...)) where the terms are the semiprimes: A001358.

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A181050 Decimal expansion of the constant 1+3/(5+7/(9+11/(13+...))), using all odd integers in this generalized continued fraction.

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