A112302 -id:A112302 - cp's OEIS frontend

A123852 Decimal expansion of (1(2(3*...)^(1/3))^(1/3))^(1/3).

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

Offset: 1

Petros Hadjicostas and Jonathan Sondow, Oct 15 2006

Cubic recurrence constant (see A123851): a cubic analog of Somos's quadratic recurrence constant A112302.

			1.156362684332269716853370322887369356513014543891888637999259598983177816...

Steven R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.

Kh. Hessami Pilehrood and T. Hessami Pilehrood, Vacca-type series for values of the generalized-Euler-constant function and its derivative, arXiv:0808.0410 [math.NT], 2008.
Kh. Hessami Pilehrood and T. Hessami Pilehrood, Vacca-type series for values of the generalized-Euler-constant function and its derivative, Journal of Integer Sequences 13 (2010), Article 10.7.3.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332(1) (2007), 292-314.
Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.

Cf. A052129, A112302, A116603, A123851, A123853, A123854.

Take[RealDigits[Product[N[n^3^-n,200], {n,400}]][[1]], 100]
RealDigits[Exp[-D[PolyLog[n, 1/3], n]/.n->0], 10, 100][[1]] (* Jean-François Alcover, Jan 28 2014 *)

prodinf(n=1, n^(1/3^n)) \\ Michel Marcus, Aug 03 2019

Equals Product_{n>=1} n^(1/3^n).

References updated by R. J. Mathar, Aug 12 2010

A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 1, 10, 478, 68248, 21809656, 13107532816, 13244650672240, 20818058883902848, 48069880140604832128, 156044927762422185270016, 687740710497308621254625536, 4000181720339888446834235653120, 29991260979682976913756629498334208

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The multinomial transform [MNL] transforms an input sequence b(n) into the output sequence a(n). Given the fact that the structure of the a(n) formulas, see the examples, lead to the multinomial coefficients A036039 the MNL transform seems to be an appropriate name for this transform. The multinomial transform is related to the exponential transform, see A274804 and the third formula. For the inverse multinomial transform [IML] see A274844.
To preserve the identity IML[MNL[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A036039 appear.
We observe that a(0) = 1 and that this term provides no information about any value of b(n), this notwithstanding we will start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the multinomial transform of a sequence. The first program uses the first formula which was found by Paul D. Hanna, see A158876, and Vladimir Kruchinin, see A215915. The second program uses properties of the e.g.f., see the sequences A158876, A213507, A244430 and A274539 and the third formula. The third program uses information about the inverse multinomial transform, see A274844.
Some MNL transform pairs are, n >= 1: A000045(n) and A244430(n-1); A000045(n+1) and A213527(n-1); A000108(n) and A213507(n-1); A000108(n-1) and A243953(n-1); A000142(n) and A158876(n-1); A000203(n) and A053529(n-1); A000110(n) and A274539(n-1); A000041(n) and A215915(n-1); A000035(n-1) and A177145(n-1); A179184(n) and A038205(n-1); A267936(n) and A000266(n-1); A267871(n) and A000090(n-1); A193356(n) and A088009(n-1).

			Some a(n) formulas, see A036039:
  a(0) = 1
  a(1) = 1*x(1)
  a(2) = 1*x(2) + 1*x(1)^2
  a(3) = 2*x(3) + 3*x(1)*x(2) + 1*x(1)^3
  a(4) = 6*x(4) + 8*x(1)*x(3) + 3*x(2)^2 + 6*x(1)^2*x(2) + 1*x(1)^4
  a(5) = 24*x(5) + 30*x(1)*x(4) + 20*x(2)*x(3) + 20*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 10*x(1)^3*x(2) + 1*x(1)^5

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, arXiv:math/0205301 [math.CO], 2002; Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274844, A274804, A036039, A001818, A001316, A215915, A008279.
Cf. A244430, A213527, A213507, A243953, A158876, A274539, A215915.
Cf. A090998, A008298, A271703, A112302, A135002, A274540.

nmax:= 13: b := proc(n): (doublefactorial(2*n-1))^2 end: a:= proc(n) option remember: if n=0 then 1 else add(((n-1)!/(n-k)!) * b(k) * a(n-k), k=1..n) fi: end: seq(a(n), n = 0..nmax); # End first MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := exp(add(b(n)*x^n/n, n = 1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n = 0..nmax); # End second MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(log(1+add(s(n)*x^n/n!, n=1..nmax)), x, nmax+1): d := proc(n): n*coeff(f, x, n) end: a(0) := 1: a(1) := b(1): s(1) := b(1): for n from 2 to nmax do s(n) := solve(d(n)-b(n), s(n)): a(n):=s(n): od: seq(a(n), n=0..nmax); # End third MNL program.

b[n_] := (2*n - 1)!!^2;
a[0] = 1; a[n_] := a[n] = Sum[((n-1)!/(n-k)!)*b[k]*a[n-k], {k, 1, n}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 17 2017 *)

a(n) = Sum_{k=1..n} ((n-1)!/(n-k)!)*b(k)*a(n-k), n >= 1 and a(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
a(n) = n!*P(n), with P(n) = (1/n)*(Sum_{k=0..n-1} b(n-k)*P(k)), n >= 1 and P(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n) with b(n) = A001818(n) = ((2*n-1)!!)^2.
denom(a(n)/2^n) = A001316(n); numer(a(n)/2^n) = [1, 1, 5, 239, 8531, 2726207, ...].

A123853 Numerators in an asymptotic expansion for the cubic recurrence sequence A123851.

Original entry on oeis.org

1, 3, -15, 113, -5397, 84813, -3267755, 74391561, -15633072909, 465681118929, -31041303829713, 1145088996404679, -185348722911971841, 8165727090278785521, -778296382754673737187, 39898888480559205453945, -35033447016186321707305533

Offset: 0

Petros Hadjicostas and Jonathan Sondow, Oct 15 2006

A cubic analog of the asymptotic expansion A116603 of Somos's quadratic recurrence sequence A052129. Denominators are A123854.

			A123851(n) ~ c^(3^n)*n^(-1/2)/(1 + 3/(4*n) - 15/(32*n^2) + 113/(128*n^3) - 5397/(2048*n^4) + ...) where c = 1.1563626843322... is the cubic recurrence constant A123852.

Steven R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.

T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161-167. [In Eq. (3.7), p. 166, the index in the summation for the Apostol-Bernoulli numbers should start at s = 0, not at s = 1. - _Petros Hadjicostas_, Aug 09 2019]
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007), 292-314.
Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.
Aimin Xu, Asymptotic expansion related to the Generalized Somos Recurrence constant, International Journal of Number Theory 15(10) (2019), 2043-2055. [The author gives recurrences and other formulas for the coefficients of the asymptotic expansion using the Apostol-Bernoulli numbers (see the reference above) and the Bell polynomials. - _Petros Hadjicostas_, Aug 09 2019]

Cf. A052129, A112302, A116603, A123851, A123852, A123854 (denominators).

f:=proc(t,x) exp(sum(ln(1+m*x)/t^m,m=1..infinity)); end; for j from 0 to 29 do numer(coeff(series(f(3,x),x=0,30),x,j)); od;

{a(n) = local(A); if(n < 0, 0, A = 1 + O(x) ; for( k = 1, n, A = truncate(A) + x * O(x^k); A += x^k * polcoeff( 3/4 * (subst(1/A, x, x^2/(1-x^2))^2/(1-x^2) - 1/subst(A, x, x^2)^(2/3)), 2*k ) ); numerator( polcoeff( A, n ) ) ) } /* Michael Somos, Aug 23 2007 */

A274181 Decimal expansion of Phi(1/2, 2, 2), where Phi is the Lerch transcendent.

Original entry on oeis.org

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

Offset: 0

Johannes W. Meijer and N. H. G. Baken, Jun 17 2016, Jul 08 2016

The exponential integral distribution is defined by p(x, m, n, mu) = ((n+mu-1)^m * x^(mu-1) / (mu-1)!) * E(x, m, n), see A163931 and the Meijer link. The moment generating function of this probability distribution function is M(a, m, n, mu) = Sum_{k>=0}(((mu+k-1)!/((mu-1)!*k!)) * ((n+mu-1) / (n+mu+k-1))^m * a^k).

In the special case that mu=1 we get p(x, m, n, mu=1) = n^m * E(x, m, n) and M(a, m, n, mu=1) = n^m * Phi(a, m, n), with Phi the Lerch transcendent. If n=1 and mu=1 we get M(a, m, n=1, mu=1) = polylog(m, a)/a = Li_m(a)/a.

			0.32896210586005002361062528063872043497679389922...

William Feller, An introduction to probability theory and its applications, Vol. 1. p. 285, 1968.

J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No. 3, April 1987. pp 209-211.
Eric W. Weisstein’s World of Mathematics, Lerch transcendent.
Eric W. Weisstein’s World of Mathematics, Polylogarithm.

Cf. A163931, A002162 (Phi(1/2, 1, 1)/2), A076788 (Phi(1/2, 2, 1)/2), A112302, A008276.

Digits := 101; c := evalf(LerchPhi(1/2, 2, 2));

N[HurwitzLerchPhi[1/2, 2, 2], 25] (* G. C. Greubel, Jun 19 2016 *)

Pi^2/3 - 2*log(2)^2 - 2 \\ Altug Alkan, Jul 08 2016

lerchphi(.5,2,2) \\ Charles R Greathouse IV, Jan 30 2025

from mpmath import mp, lerchphi
mp.dps=102
print([int(d) for d in list(str(lerchphi(1/2, 2, 2))[2:-1])]) # Indranil Ghosh, Jul 04 2017

Equals Phi(1/2, 2, 2) with Phi the Lerch transcendent.
Equals Sum_{k>=0}(1/((2+k)^2*2^k)).
Equals 4 * polylog(2, 1/2) - 2.
Equals Pi^2/3 - 2*log(2)^2 - 2.
Equals Integral_{x=0..oo} x*exp(-x)/(exp(x)-1/2) dx. - Amiram Eldar, Aug 24 2020

A296301 Decimal expansion of Product_{k>=2} k^(1/k!).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Dec 09 2017

			1.8290246795635718643895723573648858491007676333721141167306441...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Nested Radical
Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant

Cf. A112302, A115522, A123852, A188835, A259235.

RealDigits[Exp[Sum[ Log[k]/k!, {k, 2, 700}]], 10, 100][[1]] (* G. C. Greubel, Jul 28 2018 *)

exp(suminf(k=2, log(k)/k!)) \\ Michel Marcus, Dec 11 2017

Equals (2*(3*(4*(5*(6*(7*...)^(1/7))^(1/6))^(1/5))^(1/4))^(1/3))^(1/2).
Equals exp(Sum_{k>=2} log(k)/k!).
Equals lim_{k->infinity} b(k)^(1/k!), where b(k) = k*b(k-1)^k with b(0) = 1.
Equals Product_{p prime} p^(Sum_{k>=2} (p-adic valuation of k)/k!).

A188834 Decimal expansion of limit sqrt(2sqrt(4sqrt(6sqrt(8sqrt(10...sqrt(2n...)))))).

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava, Apr 12 2011

			3.3233758992671882...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A112302, A171759, A188835.

with(numtheory);
P:=proc(i)
local a,n;
a:=1;
for n from i by -1 to 1 do a:=2*n*sqrt(a); od;
print(evalf(sqrt(a),1000));
end:
P(5000);

digits = 101; p[m_] := p[m] = Fold[N[Sqrt[#2*#1], digits] &, 1, Range[2*m, 2, -2]] // RealDigits[#, 10, digits]& // First; p[digits]; p[m = 2*digits]; While[p[m] != p[m/2], m = 2*m]; p[m] (* Jean-François Alcover, Feb 24 2014 *)

A188835 Decimal expansion of limit sqrt(1sqrt(3sqrt(5sqrt(7sqrt(9...sqrt((2n-1)...)))))).

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava, Apr 12 2011

A generalization of Somos quadratic recurrence constant A112302.

			2.10784090263785493...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A112302, A171759, A188834

with(numtheory);
P:=proc(i)
local a,n;
a:=1;
for n from i by -1 to 1 do a:=(2*n-1)*sqrt(a); od;
print(evalf(sqrt(a),1000));
end:
P(5000);

digits = 100; Clear[p]; p[m_] := p[m] = Fold[N[Sqrt[#2*#1], digits] &, 1, Range[2*m + 1, 1, -2]] // RealDigits[#, 10, digits] & // First; p[digits]; p[m = 2*digits]; While[p[m] != p[m/2], m = 2*m]; p[m] (* Jean-François Alcover, Feb 24 2014 *)

Product_{n>0} ((2*n + 1) / (2*n - 1)) ^ (2^-n). - Michael Somos, Feb 24 2014

A259235 Decimal expansion of sqrt(2sqrt(3sqrt(4*...))), a variant of Somos's quadratic recurrence constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 22 2015

			2.7612068419574980332304546465801311048761259807153...

G. C. Greubel, Table of n, a(n) for n = 1..10000
StackExchange, Improving bound on sqrt(2*sqrt(3*sqrt(4*...)))
Eric Weisstein's MathWorld, Somos's Quadratic Recurrence Constant

Cf. A112302, A114124.

SetDefaultRealField(RealField(100)); Exp(2*(&+[(1/2)^n*Log(n): n in [2..2000]])); // G. C. Greubel, Sep 30 2018

RealDigits[Exp[-2*Derivative[1, 0][PolyLog][0, 1/2]], 10, 102] // First
RealDigits[Exp[2*Sum[(1/2)^n*Log[n], {n, 2, 2000}]], 10, 100][[1]] (* G. C. Greubel, Sep 30 2018 *)

exp(sumpos(n=1,log(n+1)/2^n)) \\ Charles R Greathouse IV, Apr 18 2016

Equals A112302^2.
Equals exp( Sum_{n>=1} log(n)/2^(n-1) ).
Also equals exp(-2*PolyLog'(0,1/2)), where PolyLog' is the derivative of PolyLog(n,x) w.r.t. n.

A268107 Decimal expansion of 'lambda', a Somos quadratic recurrence constant mentioned by Steven Finch.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jan 26 2016

[Quoted from Steven Finch] Another Somos constant lambda = 0.3995246670... arises as follows: If k < lambda, then the sequence h_0 = 0, h_1 = k, h_n = h_(n-1)*(1 + h_(n-1) - h_(n-2)) for n>=2 converges to a limit less than 1; if k > lambda, then the sequence diverges to infinity. This is similar to Grossman's constant.

A heuristical evaluation of lambda = 0.39952466709679946552503347433225833221736985467599... was communicated to me by Jon E. Schoenfield in a private email.

			0.39952466709679946552503347433225833221736985467599689773670052894853...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic Recurrence Constants, p. 446.

Steven R. Finch, A Deceptively Simple Quadratic Recurrence, arXiv:2409.03510 [math.NT], 2024. See p. 9.
Jon E. Schoenfield, Magma program communicated to J.-F. Alcover.
Eric Weisstein's MathWorld, Somos's Quadratic Recurrence Constant

Cf. A045761, A085835, A112302.

// See the link to Jon E. Schoenfield's program.

n0 (* initial number of terms *) = 2*10^7; iter = 10^5; dn = 10^6; k1 = 0.3; k2 = 0.4; eps = 10^-16; f[k_?NumericQ] := (h0 = 0; h1 = k; h2 = k*(1+k); Do[h0 = h1; h1 = h2; h2 = Min[h1 + (h1-h0), h1*(1+h1-h0)], {iter}]; h2); Clear[g]; g[n0] = k1; g[n = n0+dn] = k2; g[n_] := g[n] = k /. FindRoot[f[k]==1, {k, g[n-dn] }]; While[Print[n, " ", g[n] // RealDigits]; Abs[g[n] - g[n-dn]] > eps, n = n+dn]; lambda = g[n]; RealDigits[lambda][[1]][[1;;9]]

Conjecture: lambda is the radius of convergence of the function Sum_{n>=0} A045761(n)*x^n, that is the constant 1/d computed by Vaclav Kotesovec in A045761.

Extended to 127 digits using Jon E. Schoenfield's evaluation, Aug 27 2016

A380373 Decimal expansion of Sum_{i>=1} 1/2^A082851(i).

Original entry on oeis.org

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

Offset: 0

Jwalin Bhatt, Jan 23 2025

This number has the property that the geometric mean of the differences (A082850) in the positions of 1s (A082851) of its binary representation (A380372) approaches the Somos constant (A112302).

			0.8641913214950458_10 -> 0.1101110100111011101_2 (A380372)
Positions of 1s -> 1,2,4,5,6,8,11,12,13,15,... (A082851)
Difference in positions of 1s -> 1,1,2,1,1,2,3,1,1,2,1,1,2,3,4,... (A082850)
Geomtric Mean -> 1.66168794963359... (A112302)

Wikipedia, Somos Constant

Cf. A082850, A082851, A380372 (binary expansion).

from itertools import count, islice
from fractions import Fraction
import os
def A380372_gen():
    S = []
    for n in count(1):
        yield from (m:=S+[0]*(n-1)+[1])
        S += m
def bin_to_frac_interval(binary_repr):
    lower_bound, last_bit_id = 0, 0
    for i, bit in enumerate(binary_repr, start=1):
        if bit:
            lower_bound += Fraction(1, 2**i)
            last_bit_id = i
    upper_bound = lower_bound + Fraction(1, 2**last_bit_id)
    return lower_bound, upper_bound
n_binary_terms = 400
diff_bin = islice(A380372_gen(), n_binary_terms)
lower, upper = bin_to_frac_interval(diff_bin)
lower, upper = str(int(lower*10**(n_binary_terms//3))), str(int(upper*10**(n_binary_terms//3)))
A380373 = os.path.commonprefix([lower, upper])

Equals Sum_{i>=1} 1/2^A082851(i).

Equals Sum_{i>=1} A380372(i)/2^i.

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