A192918 -id:A192918 - cp's OEIS frontend

A256099 Decimal expansion of the real root of a cubic used by Omar Khayyám in a geometrical problem.

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

Offset: 1

Wolfdieter Lang, Apr 08 2015

This geometrical problem is considered in the Alten et al. reference on pp. 190-192.
The geometrical problem is to find in the first quadrant the point P on a circle (radius R) such that the ratio of the normal to the y-axis  through P and the radius equals the ratio of the segments of the radius on the y-axis. See the link with a figure and more details. For Omar Khayyám see the references as well as the Wikipedia and MacTutor Archive links.
The ratio of the length of the normal x and the segment h on the y-axis starting at the origin is called xtilde, and satisfies the cubic equation
  xtilde^3 -2*xtilde^2 + 2*xtilde - 2 = 0. This xtilde is the tangent of the angle alpha between the positive y-axis and the radius vector from the origin to the point P. This cubic equation has only one real solution xtilde = tan(alpha) given in the formula section. The present decimal expansion belongs to xtilde.
Apart from the first digit the same as A192918. - R. J. Mathar, Apr 14 2015

			1.5436890126920763615708559...

H.-W. Alten et al., 4000 Jahre Algebra, 2. Auflage, Springer, 2014, pp. 190-192.
O. Khayyam, A paper of Omar Khayyam, Scripta Math. 26 (1963), 323-337.

Wolfdieter Lang, A Geometrical Problem of Omar Khayyám and its Cubic.
MacTutor History of Mathematics archive, Omar Khayyám
Wikipedia, Omar Khayyám

Cf. A058265. Essentially the same as A192918.

RealDigits[Root[x^3 - 2 x^2 + 2 x - 2, 1], 10, 105][[1]] (* Jean-François Alcover, Oct 24 2019 *)

solve(x=1, 2, x^3-2*x^2+2*x-2) \\ Michel Marcus, Oct 24 2019

xtilde = tan(alpha) = ((3*sqrt(33) + 17)^(1/3) - (3*sqrt(33) - 17)^(1/3) + 2)/3 = 1.54368901269...
The corresponding angle alpha is approximately 57.065 degrees.
The real root of x^3-2*x^2+2*x-2. Equals tau^2-tau where tau is the tribonacci constant A058265. - N. J. A. Sloane, Jun 19 2019

A319200 a(n) = -(A(n) - A(n-1)) where A(n) = A057597(n+1), for n >= 0.

Original entry on oeis.org

0, -1, 2, -1, -2, 5, -4, -3, 12, -13, -2, 27, -38, 9, 56, -103, 56, 103, -262, 215, 150, -627, 692, 85, -1404, 2011, -522, -2893, 5426, -3055, -5264, 13745, -11536, -7473, 32754, -36817, -3410, 72981, -106388, 29997, 149372, -285757, 166382, 268747, -720886, 618521, 371112, -1710519, 1957928, 123703, -3792150

Offset: 0

Wolfdieter Lang, Oct 23 2018

This sequence appears in the reduction formula for negative powers of the tribonacci constant t = A058265: t^(-n) = A(n)*t^2 + a(n)*t + A(n+1)*1, with A(n) = A057597(n+1), for n >= 0. This follows from t^3 = t^2 + t + 1, or 1/t = t^2 - t - 1 = A192918, leading to the recurrence: A(n) = -A(n) - A(n-1) + A(n-2), with inputs A(-3) = 1, A(-2) = 1 and A(-1) = 0 and a(n) = -(A(n) - A(n-1)). See the example below.

			The coefficients of t^2, t, 1 for t^(-n) begin, for n >= -3:
n     t^2  t   1
-----------------
-3     1   1   1
-2     1   0   0
-1     0   1   0
----------------
+0     0   0   1
+1     1  -1  -1
+2    -1   2   0
+3     0  -1   2
+4     2  -2  -3
+5    -3   5   1
+6     1  -4   4
+7     4  -3  -8
+8    -8  12   5
+9     5 -13   7
10     7  -2 -20
...

Index entries for linear recurrences with constant coefficients, signature (-1,-1,1).

Cf. A057597, A058265, A078016(n+1) (different signs), A192918.

LinearRecurrence[{-1,-1,1},{0,-1,2},60] (* Harvey P. Dale, Jul 20 2025 *)

a057597(n) = polcoeff( if( n<0, x / ( 1 - x - x^2 - x^3), x^2 / ( 1 + x + x^2 - x^3) ) + x*O(x^abs(n)), abs(n)) \\ after Michael Somos in A057597
a(n) = -(a057597(n+1)-a057597(n)) \\ Felix Fröhlich, Oct 23 2018

a(n) = -(A057597(n+1) - A057597(n)), for n >= 0.
Recurrence  a(n) = -a(n-1) - a(n-2) + a(n-3), for n >=0, with a(-3) = 1, a(-2) = 0 and a(-1) = 1.
G.f.: (1 + 1/x)/(1 + x + x^2 - x^3).

A321196 Riordan triangle T = (1/(1 + x^2 - x^3), x/(1 + x^2 - x^3)).

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 1, -2, 0, 1, 1, 2, -3, 0, 1, -2, 3, 3, -4, 0, 1, 0, -6, 6, 4, -5, 0, 1, 3, -1, -12, 10, 5, -6, 0, 1, -2, 12, -4, -20, 15, 6, -7, 0, 1, -3, -7, 30, -10, -30, 21, 7, -8, 0, 1, 5, -16, -15, 60, -20, -42, 28, 8, -9, 0, 1

Offset: 0

Wolfdieter Lang, Nov 09 2018

This is the (ordinary) convolution triangle based on A077961 (the column k = 0 of T).
The row polynomials R(n, x) := Sum_{k=0..n} T(n, k)*x^k, with R(-1, x) = 0, appear in the Cayley-Hamilton formula for nonnegative powers of a 3 X 3 matrix with Det M = sigma(3; 3) = x1*x2*x3 = +1, sigma(3; 2) := x1*x2 + x1*x*3 + x2*x^3 = +1 and Tr M = sigma(3; 1) = x1 + x2 = x, where x1, x2, and x3 are the eigenvalues of M, and sigma the elementary symmetric functions, as M^n = R(n-2, x)*M^2 + (-R(n-3, x) + R(n-4, x))*M + R(n-3, x)*1_3, for n >= 3, where M^0 = 1_3 is the 3 X 3 unit matrix.
For the Cayley-Hamilton formula for 3 X 3 matrices with Det M = +1, sigma(3,2) = -1 and Tr(M) = x see A104578.
The row sums give A133872 (repeat(1, 1, 0, 0)). The alternating row sums give A057597(n+2), for n >= 0.
The Riordan triangle (1/(1 + x^2 + x^3), x/(1 + x^2 + x^3)) has entries t(n, m) = (-1)^(n-m)*T(n, m) (from the g.f. G(-x, -z), where the g.f. G of T is given below).
The inverse of Riordan T is T^{-1}, given in A321198.

			The triangle T(n, k) begins:
n\k  0   1   2   3   4   5  6  7  8  9 10 ...
---------------------------------------------
0:   1
1:   0   1
2:  -1   0   1
3:   1  -2   0   1
4:   1   2  -3   0   1
5:  -2   3   3  -4   0   1
6:   0  -6   6   4  -5   0  1
7:   3  -1 -12  10   5  -6  0  1
8:  -2  12  -4 -20  15   6 -7  0  1
9:  -3  -7  30 -10 -30  21  7 -8  0  1
10:  5 -16 -15  60 -20 -42 28  8 -9  0  1
...
Cayley-Hamilton formula for the matrix TS(x) =[[x,-1,1], [1,0,0], [0,1,0]] with Det(TS(x)) = +1, sigma(3, 2) = +1, and Tr(TS(x)) = x. For n = 3: TS(x)^3 = R(1, x)*TS(x)^2 + (-R(0, x) + R(-1, x))*TS(x) + R(0, x)*1_3 = x*TS(x)^2 - TS(x) + 1_3. Compare this for x = -1 with r^3 = R(3)*r^2 + (-R(2) + R(1))*r + R(2)*1 = r^2 - r + 1, where r = 1/t = A192918, with the tribonacci constant t = A058265, and R(n) = A057597(n) = R(n-2, -1).
Recurrence: T(5, 2) = T(4, 1) - T(3, 2) + T(2, 2) = 1 -(-1) + 1 = 3.
Boas-Buck type recurrence with B = {0, -2, 3, ...}:
  T(5, 2) = ((2+1)/(5-2))*(3*1 + (-2)*0 + 0*(-3)) = 1*3 = 3.
Z- and A-recurrence with A(n) = {1, 0, -1, 1, -1, ...} and Z(n) = A(n+1):
  T(4, 0) = 0*T(3, 0) - 1*T(3, 1) + 1*T(3, 2) - 1*T(3, 3) = 0 + 2 + 0 - 1 = 1.
  T(5, 2) = 1*T(4, 1) + 0*T(4, 2) - 1*T(4, 3) + 1*T(4, 4) = 2 + 0 + 0 + 1 = 3.

Cf. A057597, A058265, A077961, A001005, A104578, A112455, A133872, A192918, A321197, A321198.

T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] - T[n - 2, k] + T[n - 3, k]; T[0, 0] = 1; T[, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jul 06 2019 *)

# uses[riordan_array from A256893]
riordan_array(1/(1 + x^2 - x^3), x/(1 + x^2 - x^3), 11) # Peter Luschny, Nov 13 2018

T(n, k) = T(n-1, k-1) - T(n-2, k) + T(n-3, k), T(0, 0) = 1, T(n,k) = 0 if n < k or if k < 0. (Cf. A104578.)
The Riordan property T = (G(x), x*G(x)) with G(x) = 1/(1 + x^2 - x^3) implies the following.
G.f. of row polynomials R(n, x) is G(x, z) = 1/(1 - x*z + z^2 - z^3).
G.f. of column sequence k: x^k/(1 + x^2 - x^3)^(k+1), k >= 0.
Boas-Buck recurrence (see the Aug 10 2017 remark in A046521, also for two references):
T(n, k) = ((k+1)/(n-k))*Sum_{j=k..n-1} B(n-1-j)*T(j, k), for n >= 1, k = 0,1, ..., n-1, and input T(n, n) = 1, for n >= 0. Here B(n) = [x^n]*(d/dx)log(G(x)) = x*(-2 + 3*x)/(1 + x^2 - x^3) = (-1)^n*A112455(n+1), for n >= 0.
Recurrences from the A- and Z- sequences (see the W. Lang link under A006232 with references), which are A(n) = A321197(n) and Z(n) = A(n+1).
  T(0, 0) = 1, T(n, k) = 0 for n < k, and
  T(n, 0) = Sum_{j=0..n-1} Z(j)*T(n-1, j), for n >= 1, and
  T(n, k) = Sum_{j=0..n-k} A(j)*T(n-1, k-1+j), for n >= m >= 1.

A376841 Decimal expansion of a constant related to the asymptotics of A066447 and A333374.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Oct 06 2024

			7.1578741786143524880205016499891016064826797593549373619575862725233...

Cf. A066447, A192918, A333374.

RealDigits[E^(2*Sqrt[Log[r]^2 + PolyLog[2, r^2] - PolyLog[2, -r^2]]) /. r -> (-1 - 2/(17 + 3*Sqrt[33])^(1/3) + (17 + 3*Sqrt[33])^(1/3))/3, 10, 105][[1]]

Equals limit_{n->infinity} A066447(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A333374(n)^(1/sqrt(n)).
Equals exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r.

A376853 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

Original entry on oeis.org

1, 1, 4, 9, 16, 28, 49, 84, 140, 228, 361, 560, 856, 1288, 1916, 2821, 4108, 5928, 8480, 12024, 16920, 23637, 32788, 45196, 61928, 84368, 114332, 154160, 206857, 276308, 367476, 486680, 641996, 843656, 1104592, 1441168, 1873965, 2428816, 3138132, 4042408, 5192132

Offset: 0

Vaclav Kotesovec, Oct 06 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001935, A143184, A192918, A376812, A376852, A376854.

nmax = 60; CoefficientList[Series[Sum[x^(k*(k+1)/2) * Product[(1+x^j)/(1-x^j), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

a(n) ~ c * exp(sqrt(8*n*(log(r)^2 + polylog(2,r) - polylog(2,-r)))), where r = A192918 = 0.54368901269207636157... is the real root of the equation r*(1+r^2) = (1-r^2) and c = 0.0643033662740307713580663125340126524175...

A273905 Number of symmetric bargraphs having semiperimeter n (n>=2).

Original entry on oeis.org

1, 2, 3, 5, 9, 15, 27, 46, 83, 143, 259, 450, 817, 1429, 2599, 4570, 8323, 14698, 26797, 47491, 86659, 154042, 281287, 501283, 915907, 1635835, 2990383, 5351138, 9786369, 17541671, 32092959, 57610988, 105435607, 189521640, 346950321, 624389105

Offset: 2

Emeric Deutsch and Sergi Elizalde, Jun 23 2016

nonn

			a(4) = 3; indeed, the corresponding compositions are [3],[2,2],[1,1,1].
a(6) = 9; indeed, the corresponding compositions are [5],[4,4],[1,3,1],[2,3,2],[2,1,2],[3,3,3],[2,2,2,2],[1,2,2,1],[1,1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 2..2000
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Q := sqrt((1-z^2)*(1-z-z^2-z^3)*(1+z-z^2+z^3)): g := (1/2)*(1+z)*(z^4+2*z^3+2*z^2-1+Q)/(z*(1-z-z^2-z^3)): gser := series(g, z = 0, 42): seq(coeff(gser,z,n), n=2..37);
# second Maple program:
a:= proc(n) option remember; `if`(n<9, [$0..3, 5, 9, 15, 27]
      [n], (2*a(n-1) +(4*n-6)*a(n-2) -(2*n-12)*a(n-4)
      -6*a(n-5) +2*a(n-6) -(n-9)*a(n-8))/ (n+1))
    end:
seq(a(n), n=2..40);  # Alois P. Heinz, Jun 24 2016

a[2]=1; a[3]=2; a[4]=3; a[5]=5; a[6]=9; a[7]=15; a[8]=27; a[n_ /; n>8] := a[n] = ((9-n)*a[n-8] + 2*a[n-6] - 6*a[n-5] + (12-2*n)*a[n-4] + (4*n-6)*a[n-2] + 2*a[n-1])/(n+1); Table[a[n], {n, 2, 40}] (* Jean-François Alcover, Dec 02 2016, after Alois P. Heinz *)

G.f.: g(z)=(1+z)(z^4+2z^3+2z^2-1+Q)/(2z(1-z-z^2-z^3)), where Q = sqrt((1-z^2)(1-z-z^2-z^3)(1+z-z^2+z^3)).
Conjecture D-finite with recurrence (n+1)*a(n) +2*(-1)*a(n-1) +2*(-2*n+3)*a(n-2) +2*(n-6)*a(n-4) +6*(1)*a(n-5) -2*a(n-6) +(n-9)*a(n-8)=0. - R. J. Mathar, Jul 22 2022
a(n) ~ sqrt(2*r*(2-3*r)) * (25 + 18*r + 13*r^2) * (1 + r + r^2)^n / (22*sqrt(Pi*n)), where r = A192918. - Vaclav Kotesovec, Mar 08 2023

A309172 Expansion of Product_{k>=1} 1/(1 - (1 + x + x^2) * x^k).

Original entry on oeis.org

1, 1, 3, 7, 15, 31, 64, 128, 254, 496, 961, 1844, 3516, 6662, 12564, 23593, 44153, 82385, 153351, 284857, 528235, 978148, 1809120, 3342722, 6171318, 11385733, 20994298, 38693809, 71288111, 131297855, 241761727, 445068646, 819205061, 1507641487, 2774307387, 5104712633

Offset: 0

Ilya Gutkovskiy, Jul 15 2019

nonn

Cf. A160571, A227681, A309173.

nmax = 35; CoefficientList[Series[Product[1/(1 - (1 + x + x^2) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[Exp[Sum[x^k Sum[(1 + x + x^2)^d/d, {d, Divisors[k]}], {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (1 + x + x^2)^d/d).

a(n) ~ 1/((1 + 2*r + 3*r^2) * QPochhammer[r] * r^(n+1)), where r = A192918. - Vaclav Kotesovec, Jul 16 2019

A320286 Expansion of Product_{k>=1} 1/(1 - x^k - x^(2k) - x^(3k)).

Original entry on oeis.org

1, 1, 3, 6, 13, 24, 51, 93, 184, 343, 654, 1211, 2286, 4217, 7865, 14521, 26912, 49600, 91669, 168800, 311305, 573058, 1055576, 1942437, 3575840, 6578762, 12106121, 22270404, 40972700, 75367724, 138644224, 255020102, 469095029, 862827347, 1587061299, 2919111935, 5369224903

Offset: 0

Ilya Gutkovskiy, Oct 09 2018

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000010, A001935, A082303, A093305, A162891.

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/( &*[(1-x^k-x^(2*k)-x^(3*k)): k in [1..m+2]]))); // G. C. Greubel, Oct 24 2018

seq(coeff(series(mul(((1-x^k-x^(2*k)-x^(3*k)))^(-1),k=1..n),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 25 2018

nmax = 36; CoefficientList[Series[Product[1/(1 - x^k - x^(2 k) - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 36; CoefficientList[Series[Exp[-Sum[Sum[EulerPhi[j] Log[1 - x^(j k) (1 + x^(j k) + x^(2 j k))]/(j k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]

m=40; x='x+O('x^m); Vec(1/prod(k=1, m+2, (1-x^k-x^(2*k)-x^(3*k)))) \\ G. C. Greubel, Oct 24 2018

G.f.: exp(-Sum_{k>=1} Sum_{j>=1} phi(j)*log(1 - x^(j*k)*(1 + x^(j*k) + x^(2*j*k)))/(j*k)), where phi = Euler totient function (A000010).
From Vaclav Kotesovec, Oct 09 2018: (Start)
a(n) ~ s*p / r^(n+1), where
r = A192918 = ((17 + 3*sqrt(33))^(1/3) - 2/(17 + 3*sqrt(33))^(1/3) - 1)/3 = 0.54368901269207636157085597180174798652520329765098393524... is the real root of the equation 1 - r - r^2 - r^3 = 0,
s = (51 + 9*sqrt(33))/(4*(17 + 3*sqrt(33))^(1/3) + (17 + 3*sqrt(33))^(5/3) - 34 - 6*sqrt(33)) = 0.3362281169949410942253629540143324151579260900204592... is the real root of the equation -1 - 2*s + 44*s^3 = 0,
p = Product_{k>=2} 1/(1 - r^k - r^(2*k) - r^(3*k)) = 2.577933056783997593784130068093034525002002622982961271582417329674...
(End)

A347290 Arnoux-Rauzy word sigma_0 x sigma_2 x sigma_1. Fixed point of the morphism 0-> 0201020, 1->1020, 2->201020 starting from a(1)=0.

Original entry on oeis.org

0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 0

Offset: 1

Jiri Hladky, Aug 29 2021

nonn

Arnoux-Rauzy word sigma_0 x sigma_2 x sigma_1, where sigmas are defined as:
  sigma_0 : 0 -> 0, 1 -> 10, 2 -> 20;
  sigma_1 : 0 -> 01, 1 -> 1, 2 -> 21;
  sigma_2 : 0 -> 02, 1 -> 12, 2 -> 2.
Fixed point of the morphism 0->0201020, 1->1020, 2->201020 starting from a(1)=0.
Frequency of letters:
  0: 1/t   ~ 54.368% (A192918)
  1: 1/t^3 ~ 16.071%
  2: 1/t^2 ~ 29.559%
where t is tribonacci constant A058265.
Equals A286998 with a re-mapping of values 1->2, 2->1.

L. Balková, M. Bucci, A. De Luca, J. Hladký and S. Puzynina, Aperiodic Pseudorandom Number Generators Based on Infinite Words, Theoret. Comput. Sci. 647 (2016), 85-100.
Julien Cassaigne, Sebastien Ferenczi, and Luca Q. Zamboni, Imbalances in Arnoux-Rauzy sequences, Annales de l'institut Fourier, 50 (2000), 1265-1276.
D. Damanik and L. Q. Zamboni, Arnoux-Rauzy subshifts: linear recurrence, powers and palindromes, arXiv:math/0208137 [math.CO], 2002.
J. Patera, Generating the Fibonacci chain in O(log n) space and O(n) time (2003)
Gérard Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110.2 (1982): 147-178.
Index entries for sequences that are fixed points of mappings

Cf. A080843 A286998 (values 0,2,1), A058265.

Nest[ Flatten[# /. {0 -> {0, 2, 0, 1, 0, 2, 0}, 1 -> {1, 0, 2, 0}, 2 -> {2, 0, 1, 0, 2, 0}}] &, {0}, 3]

A376152 Decimal expansion of a constant related to the asymptotics of A376530.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Oct 09 2024

			4.988020876600903805335224460790773050832038156091687962387444991919...

Cf. A376530, A376621.

RealDigits[E^(2*Sqrt[Log[r]^2 + 2*PolyLog[2, 1-r] - 2*PolyLog[2, 1-r^3]/3]) /. r -> (-1 - 2/(17 + 3*Sqrt[33])^(1/3) + (17 + 3*Sqrt[33])^(1/3))/3, 10, 120][[1]]

Equals limit_{n->infinity} A376530(n)^(1/sqrt(n)).

Equals exp(2*sqrt(log(r)^2 + 2*polylog(2, 1-r) - 2*polylog(2, 1-r^3)/3)), where r = A192918 = 0.54368901269207636157085597180174... is the real root of the equation r^2 * (1-r^3)^2 = (1-r)^2.

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