A073007 -id:A073007 - cp's OEIS frontend

A072558 Decimal expansion of the one-ninth constant.

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

Offset: 0

Robert G. Wilson v, Aug 03 2002

The generating function of A113184 equals 1/8 at q = Lambda = 0.1076539192... where K(k)=2E(k). - Michael Somos, Jul 21 2006

			0.1076539192264845766153234450909471905879...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 259-262.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Steven R. Finch, The "One-Ninth" Constant [Broken link]
Steven R. Finch, The "One-Ninth" Constant [From the Wayback machine]
Alphonse P. Magnus, Jean Meinguet, The elliptic functions and integrals of the '1/9' problem
Alphonse P. Magnus, Jean Meinguet, The elliptic functions and integrals of the '1/9' problem, presented at Antwerpen international conference on rational approximation, 1999, ICRA99, Numerical Algorithms 24: (1-2) (2000) 117-139.
Simon Plouffe, The One-ninth constant
Eric Weisstein's World of Mathematics, One-Ninth Constant

Cf. A072559, A073007.

c = k /. FindRoot[ EllipticK[k^2] == 2*EllipticE[k^2], {k, 9/10}, WorkingPrecision -> 120]; Take[ RealDigits[ N[Exp[-Pi*(EllipticK[1 - c^2] / EllipticK[c^2])], 120]][[1]], 105] (* Jean-François Alcover, Jul 28 2011, after MathWorld *)
RealDigits[q /. FindRoot[4 EllipticE[InverseEllipticNomeQ[q]] == Pi EllipticTheta[3, 0, q]^2, {q, 1/9, 0, 1}, WorkingPrecision -> 105]][[1]] (* Jan Mangaldan, Jun 25 2020 *)

c=solve(x=.9,.91, ellK(x)-2*ellE(x)); exp(-Pi*ellK(sqrt(1 - c^2))/ellK(c)) \\ Charles R Greathouse IV, Feb 04 2025

A193219 Expansion of sqrt((2/Pi)*elliptic_E(k)) in powers of q.

Original entry on oeis.org

1, -2, 8, -16, 18, -32, 112, -192, 0, 62, 1840, -3312, -8320, 16480, 71840, -137280, -522174, 1011392, 4107960, -7945008, -32457600, 62909120, 261338416, -506930112, -2129035776, 4133297534, 17531850576, -34058050240, -145663683072, 283125653280, 1219649036576, -2371704375168, -10281070960128, 20000146662464, 87178011852896

Offset: 0

Joerg Arndt, Aug 26 2011

sign

Let s = 16*q*(E1*E4^2/E2^3)^8 where Ek = Product_{n>=1} (1-q^(k*n)) (s=k^2 where k is elliptic k), then the g.f. is sqrt(hypergeom([-1/2, +1/2], [+1], s)) (expansion of sqrt((2/Pi)*elliptic_E(k)) in powers of q).

The corresponding sequence for sqrt((2/Pi)*elliptic_K(k)) is A000122.

			sqrt(E(k(q))) = 1 - 2*q + 8*q^2 - 16*q^3 + 18*q^4 - 32*q^5 + 112*q^6 - 192*q^7 +- ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A194094 (elliptic_E(k(q))), A004018 (elliptic_K(k(q))), A000122 (sqrt(elliptic_K(k(q)))=Theta3(q)), A115977 (elliptic k(q)^2).

Cf. A072558, A073007.

CoefficientList[Series[Sqrt[(2/Pi) EllipticE[InverseEllipticNomeQ[q]]], {q, 0, 50}], q] (* Jan Mangaldan, Dec 07 2021 *)
nmax = 30; dtheta = D[Normal[Series[EllipticTheta[3, 0, x], {x, 0, nmax}]], x]; CoefficientList[Series[Sqrt[(EllipticTheta[4, 0, x]^4*EllipticTheta[3, 0, x] + 4*x*dtheta)/EllipticTheta[3, 0, x]^3], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 15 2023 *)

From Vaclav Kotesovec, Nov 16 2023: (Start)
abs(a(n)) ~ c * d^n / n^(3/2), where
d = 1/sqrt(A072558) = sqrt(A073007) = 3.0477902637682959365706804198489438625220426001497960504423261561153885844...
c = 0.60315114232684465914106139794838284733424313832900503234838172483814652... if n is even and
c = 0.38688142678580145044658710898009855553630625532976316366806686926256857... if n is odd. (End)

A072559 Continued fraction expansion of the One-ninth constant (A072558).

Original entry on oeis.org

0, 9, 3, 2, 5, 1, 2, 1, 3, 2, 1, 3, 1, 1, 1, 1, 11, 1, 1, 3, 1, 1, 8, 2, 3330, 1, 1, 5, 3, 7, 23, 1, 3, 3, 1, 1, 1, 1, 1, 4, 1, 5, 14, 4, 2, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 4, 4, 1, 2, 1, 1, 1, 76, 3, 1, 16, 2, 2, 1, 7, 1, 11, 1, 1, 1, 1, 1, 21, 1, 109, 2, 4, 1, 3, 6, 5, 7, 5, 5, 4, 1, 4, 2, 1, 6, 2, 1

Offset: 0

Robert G. Wilson v, Aug 05 2002

Essentially same as the continued fraction for the Varga constant (A073007).

Cf. A072558 (decimal expansion).

Offset changed by Andrew Howroyd, Jul 06 2024

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A072558 Decimal expansion of the one-ninth constant.

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A193219 Expansion of sqrt((2/Pi)*elliptic_E(k)) in powers of q.

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A072559 Continued fraction expansion of the One-ninth constant (A072558).

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