A111129 -id:A111129 - cp's OEIS frontend

A111188 Ordinary continued fraction expansion of the continued fraction 1+1/(1+2/(1+3/(1+4/(1+5/(1+...)))))... (see A111129 for more information).

1, 1, 1, 9, 2, 4, 6, 1, 7, 1, 14, 3, 2, 1, 3, 1, 7, 3, 1, 2, 119, 6, 3, 5, 1, 15, 3, 5, 1, 4, 3, 1, 1, 4, 2, 2, 3, 2, 1, 7, 1, 2, 19, 1, 3, 1, 5, 7, 4, 1, 7, 1, 12, 1, 8, 3, 1, 4, 2, 3, 5, 7, 1, 1, 2, 2, 1, 1, 122, 1, 9, 1, 4, 44, 2, 1, 16, 6, 8, 44, 1, 1, 4, 1, 7, 1, 2, 1, 3, 2, 58, 1, 6

Offset: 0

Hans Havermann, Oct 19 2005

Cf. A111129 (decimal expansion).

ContinuedFraction[Sqrt[2/(Pi*E)]/Erfc[1/Sqrt[2]], 126]

Definition corrected by Steven Finch, Feb 07 2009

Offset changed by Andrew Howroyd, Aug 03 2024

A180048 Coefficient triangle of the denominators of the (n-th convergents to) the continued fraction 1/(w+2/(w+3/(w+4/... . Conjectured to equal unsigned version of A137286.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 5, 0, 1, 8, 0, 9, 0, 1, 0, 33, 0, 14, 0, 1, 48, 0, 87, 0, 20, 0, 1, 0, 279, 0, 185, 0, 27, 0, 1, 384, 0, 975, 0, 345, 0, 35, 0, 1, 0, 2895, 0, 2640, 0, 588, 0, 44, 0, 1, 3840, 0, 12645, 0, 6090, 0, 938, 0, 54, 0, 1, 0, 35685, 0, 41685, 0, 12558, 0, 1422, 0, 65

Offset: 0

Wouter Meeussen, Aug 08 2010

Equivalence to the recurrence formula needs formal proof. This continued fraction converges to 0.525135276160981... for w=1. A conjecture by Ramanujan puts this equal to -1 + 1/(sqrt(e*Pi/2) - Sum_{k>=1} 1/(2k-1)!!).
From Alexander Kreinin, Oct 26 2015: (Start)
Let us denote the continued fraction by U(w).
Then it is easy to show that Mill's ratio, R(w) = (1 - Phi(w))/f(w), where Phi is the standard normal distribution function and f is the standard normal density function, satisfies R(w) = 1/(w + U(w)).
Indeed, R(w) = 1/(w+1/(w+2/(w+3/(w+... Then we find U(w) = 1/R(w) - w. It was proved in Alexander Kreinin (arXiv:1405.5852) that R(w+t) + Q(w, t) = exp(w*t + w^2/2)*R(t), where Q(w,t) = Sum_{k>=0} Sum_{m=0..k} q(k,m) * t^m * w^(k+1)/(k+1)!.
Substituting t=0, we obtain R(w) = exp(w^2/2)*sqrt(Pi/2) - Sum_{n>=0} w^(2n+1)/(2n+1)!!. If w=1 we obtain Ramanujan's formula. (End)

			The denominator of 1/(w+2/(w+3/(w+4/(w+5/(w+6/w))))) equals 48 + 87w^2 + 20w^4 + w^6.
From _Joerg Arndt_, Apr 20 2013: (Start)
Triangle begins
     1;
     0,     1;
     2,     0,     1;
     0,     5,     0,     1;
     8,     0,     9,     0,    1;
     0,    33,     0,    14,    0,   1;
    48,     0,    87,     0,   20,   0,   1;
     0,   279,     0,   185,    0,  27,   0,  1;
   384,     0,   975,     0,  345,   0,  35,  0,  1;
     0,  2895,     0,  2640,    0, 588,   0, 44,  0, 1;
  3840,     0, 12645,     0, 6090,   0, 938,  0, 54, 0, 1;
     0, 35685,     0, 41685,    0, ... (End)

G. C. Greubel, Table of n, a(n) for the first 75 rows, flattened
Authors?, Hungarian discussion forum
Alexander Kreinin, Combinatorial Properties of Mills' Ratio, arXiv:1405.5852 [math.CO], 2014. See Table 3.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, preprint, 2016.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences 19 (2016), Article 16.6.2.

Cf. A137286, A084950, A180047, A180049.

Cf. A111129.

Table[ CoefficientList[ Denominator[ Together[ Fold[ #2/(w+#1) &, Infinity, Reverse @ Table[ k, {k, 1, n} ] ] ] ], w ], {n, 16} ] (* or equivalently *) Clear[ p ];p[ 0 ]=1; p[ 1 ]=w; p[ n_ ]:=p[ n ]= w*p[ n-1 ] + n*p[ n-2 ]; Table[ CoefficientList[ p[ k ]//Expand, w ], {k,0,15} ]

p(0)=1; p(1)=w; p(n) = w*p(n-1) + n*p(n-2) (conjecture).
T(n,k) = T(n-1,k-1) + n*T(n-2,k), T(0,0) = 1, T(1,0) = 0, T(1,1) = 1. - Philippe Deléham, Oct 28 2013
sum_{k=0..n} T(n,k) = A000932(n). - Philippe Deléham, Oct 28 2013
T(2n,0) = A000165(n); T(2n+1,1) = A129890(n); T(2n+2,2) = A035101(n+2). - Philippe Deléham, Oct 28 2013

A108088 Decimal expansion of 1/(1+1/(1+2/(1+3/(1+4/(1+5/(1+...)))))).

Original entry on oeis.org

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

Offset: 0

Philippe Deléham, Jun 21 2005

Term of Ramanujan's formula (see A059444 and A060196).

			0.6556795424187984715438712307308112833992823328704...

S. R. Finch, "Mathematical Constants", Cambridge, pp. 423-428.

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A111129.

RealDigits[Sqrt[Pi*E/2]*Erfc[1/Sqrt[2]], 10, 111][[1]]

sqrt(Pi*exp(1)/2)*erfc(1/sqrt(2)) \\ G. C. Greubel, Feb 03 2017

Equals sqrt(Pi*e/2)*erfc(1/sqrt(2)), where erfc is the complementary error function. - Daniel Forgues, Apr 14 2011

Also equals Integral_{-infinity..infinity} (1/sqrt(2*Pi))*exp(-x^2/2)/(1+x^2) dx, where the integrand is normal PDF times Cauchy PDF. - Jean-François Alcover, Apr 28 2015

A225435 Numerators of convergents to the general continued fraction 1/(1 + 2/(1 + 3/(1 + 4/(1+ ...)))).

Original entry on oeis.org

1, 1, 2, 4, 7, 19, 68, 44, 416, 758, 6092, 24284, 10348, 110864, 997828, 4545476, 827252, 5166356, 255994804, 1289266004, 3332578444, 8757252244, 3766552348, 27079574548, 1434303566956, 4061479240156, 46849154788124, 54858398447372, 816458740546228, 189647639388428

Offset: 1

Eric W. Weisstein, May 07 2013

			1, 1/3, 2/3, 4/9, 7/12, 19/39, ... = A225435(n)/A225436(n).

Seiichi Manyama, Table of n, a(n) for n = 1..843
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Cf. A225436 (denominators).
Cf. A111129 (decimal digits of infinite c.f.).
Related to A000932.

Numerator[Table[ContinuedFractionK[k, 1, {k, 1, n}], {n, 30}]]

E.g.f.: (1/2)*(-2+e^((1/2)*z*(2+z))*(1+z)(2+sqrt(2*e*Pi)*erf(1/sqrt(2)))-e^((1/2)*(1+z)^2)*sqrt(2*Pi)*(1+z)*erf((1+z)/sqrt(2))).

Lim_{n->infinity} A225435(n)/A225436(n) = sqrt(2/(e*Pi))/erfc(1/sqrt(2))-1 = A111129.

A225436 Denominators of convergents to the general continued fraction 1/(1 + 2/(1 + 3/(1 + 4/(1+ ...)))).

Original entry on oeis.org

1, 3, 3, 9, 12, 39, 123, 87, 771, 1473, 11427, 46779, 19533, 212559, 1890093, 8691981, 1570137, 9863961, 486463449, 2459255649, 6337494039, 16694653089, 7166066763, 51605000913, 2729643372111, 7738039298811, 89176449644619, 104501330075607, 1554311845035993, 361227369257943

Offset: 1

Eric W. Weisstein, May 07 2013

1

			1, 1/3, 2/3, 4/9, 7/12, 19/39, ... = A225435(n)/A225436(n).

Seiichi Manyama, Table of n, a(n) for n = 1..843
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Cf. A225435 (numerators).
Cf. A111129 (decimal digits of infinite c.f.).
Related to A000932.

Denominator[Table[ContinuedFractionK[k, 1, {k, 1, n}], {n, 30}]]

E.g.f: (1/2)*(2+e^((1/2)*(1+z)^2)*sqrt(2*Pi)*(1+z)*(-erf(1/sqrt(2))+erf((1+z)/sqrt(2)))).

Limit_{n->oo} A225435(n)/a(n) = sqrt(2/(e*Pi))/erfc(1/sqrt(2))-1 = A111129.

A005831 a(n+1) = a(n) * (a(n-1) + 1).

Original entry on oeis.org

0, 1, 1, 2, 4, 12, 60, 780, 47580, 37159980, 1768109008380, 65702897157329640780, 116169884340604934905464739377180, 7632697963609645128663145969343357330533515068777580, 886689639639303288926299195509965193299034793881606681727875910370940270908216401980

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

A discrete analog of the derivative of t(x) = tetration base e, since t'(x) = t(x) * t(x-1) * t(x-2) * ... y = y * exp(y) * exp(exp(y)) * ... * t(x) This sequence satisfies almost the same equation but the derivative is replaced by a difference, comparable to the relations between differential equations and their associated difference equations. - Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008

			a(5) = 12 since 12 = 1*2*4 + 4.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..19
M. E. Mays, Iterating the division algorithm, Fib. Quart., 25 (1987), 204-213.

Cf. A007807.

Cf. A000058, A001622, A001113, A102575, A096436, A111129.

a005831 n = a005831_list !! n
a005831_list = 0:1:zipWith (*) (tail a005831_list) (map succ a005831_list)
-- Reinhard Zumkeller, Mar 19 2011

a=0;b=1;lst={a,b};Do[c=a*b+b;AppendTo[lst,c];a=b;b=c,{n,18}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 13 2009 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-1](a[n-2]+1)},a,{n,15}] (* Harvey P. Dale, Aug 17 2013 *)

a(0) = a(1) = 1, a(2) = 2; a(n) = a(n-1)*a(n-2)*a(n-3)*... + a(n-1). - Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008

The sequence grows like a doubly exponential function, similar to Sylvester's sequence. In fact we have the asymptotic form : a(n) ~ e ^ (Phi ^ n) where e and Phi are the best possible constants. - Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A141435 a(1) = 1, a(2) = 2; a(n) = a(n-a(1)) + a(n-a(2)) + a(n-a(3)) + a(n-a(4)) + ...

Original entry on oeis.org

1, 2, 3, 6, 11, 20, 38, 71, 132, 247, 461, 861, 1609, 3005, 5613, 10485, 19584, 36581, 68330, 127632, 238404, 445314, 831798, 1553712, 2902170, 5420945, 10125754, 18913838, 35329048, 65990929, 123264078, 230244265, 430071949, 803328933

Offset: 1

Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008

Thus we get a self-reference sequence that grows exponentially. a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-6) + a(n-11) + a(n-20) + ...
A Fibonacci-like sequence, even closer to the tribonacci numbers.
Lim n-> oo log (a(n))/n converges.

			a(6) = 20 because 20 = a(5) + a(4) + a(3) = 11 + 6 + 3
a(8) = 71 because 71 = a(7) + a(6) + a(5) + a(2) = 38 + 20 + 11 + 2

Cf. A058265, A008937, A001590, A000213, A000073, A096436, A102575, A111129.

A141435 := proc(n) option remember; local a,i; if n <= 3 then RETURN(n); else a :=0 ; for i from 1 to n-1 do if n-procname(i) < 1 then RETURN(a); else a := a+procname(n-procname(i)) ; fi; od; RETURN(a); fi; end: for n from 1 to 80 do printf("%d,",A141435(n)) ; od: # R. J. Mathar, Nov 03 2008

def A141435(terms):
    seq = [1, 2]
    for n in range(3, terms):
        s = 0
        for m in seq:
            if (n - m) > 0:
                s += seq[n - m - 1] #fix for python indexing
        seq.append(s)
    return seq
print(A141435(40)) # Andres Cruz y Corro A, Jun 19 2019

More terms from R. J. Mathar, Nov 03 2008

A246822 Decimal expansion of the expected value of the function max(x-1,0) with respect to the normal distribution (with zero mean and unit standard deviation).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 04 2014

			0.083315470587686298383062738567598577306584937464...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 52.

Cf. A111129, A240717.

Join[{0}, RealDigits[1/Sqrt[2*E*Pi] - (1/2)*Erfc[1/Sqrt[2]], 10, 102] // First]

1/sqrt(2*exp(1)*Pi) - (1/2)*erfc(1/sqrt(2)) \\ Michel Marcus, Sep 04 2014

1/sqrt(2*e*Pi) - (1/2)*erfc(1/sqrt(2)).

Also equals A240717*(1 - 1/A111129).

cp's OEIS Frontend

A111188 Ordinary continued fraction expansion of the continued fraction 1+1/(1+2/(1+3/(1+4/(1+5/(1+...)))))... (see A111129 for more information).

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A180048 Coefficient triangle of the denominators of the (n-th convergents to) the continued fraction 1/(w+2/(w+3/(w+4/... . Conjectured to equal unsigned version of A137286.

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A108088 Decimal expansion of 1/(1+1/(1+2/(1+3/(1+4/(1+5/(1+...)))))).

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A225435 Numerators of convergents to the general continued fraction 1/(1 + 2/(1 + 3/(1 + 4/(1+ ...)))).

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A225436 Denominators of convergents to the general continued fraction 1/(1 + 2/(1 + 3/(1 + 4/(1+ ...)))).

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A005831 a(n+1) = a(n) * (a(n-1) + 1).

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A141435 a(1) = 1, a(2) = 2; a(n) = a(n-a(1)) + a(n-a(2)) + a(n-a(3)) + a(n-a(4)) + ...

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A246822 Decimal expansion of the expected value of the function max(x-1,0) with respect to the normal distribution (with zero mean and unit standard deviation).