A072558 -id:A072558 - cp's OEIS frontend

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

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

A113184 Absolute difference between sum of odd divisors of n and sum of even divisors of n.

Original entry on oeis.org

1, 1, 4, 5, 6, 4, 8, 13, 13, 6, 12, 20, 14, 8, 24, 29, 18, 13, 20, 30, 32, 12, 24, 52, 31, 14, 40, 40, 30, 24, 32, 61, 48, 18, 48, 65, 38, 20, 56, 78, 42, 32, 44, 60, 78, 24, 48, 116, 57, 31, 72, 70, 54, 40, 72, 104, 80, 30, 60, 120, 62, 32, 104, 125, 84, 48, 68, 90, 96, 48, 72

Offset: 1

Michael Somos, Oct 17 2005

The generating function equals 1/8 at q = Lambda = 0.1076539192... (A072558) the "One-Ninth" constant. - Michael Somos, Jul 21 2006
Absolute value of A002129. - John W. Layman, Sep 27 2012
The Möbius transform is 1, 0, 3, 4, 5, 0, 7, 8, 9, 0, 11, 12, 13, 0, 15, 16, 17, 0, 19, 20, 21, 0, 23, 24, 25, 0, 27, ... - R. J. Mathar, Jan 08 2013

			From _Peter Bala_, Dec 11 2020: (Start)
n = 15: n is a triangular number, so e(n) = (-1)^(n+1)*n = 15 and a(15) = 15 + a(14) + a(12) - a(9) - a(5) =  15 + 8 + 20 - 13 - 6 = 24;
n = 16: n is a not triangular number, so e(n) = 0 and a(16) = a(15) + a(13) - a(10) - a(6) + a(1) =  24 + 14 - 6 - 4 + 1 = 29. (End)

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, One-Ninth Constant.

Cf. A002129(n) = -(-1)^n a(n).

Cf. A072558, A284362.

f[n_]:=Module[{dn=Divisors[n],odn,edn},odn=Select[dn,OddQ];edn=Select[dn,EvenQ];Abs[Total[odn]-Total[edn]]]
f/@Range[80]  (* Harvey P. Dale, Feb 25 2011 *)
max = 80; s = (1/x)*Sum[k*x^k/(1 - (-x)^k), {k, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 04 2015 *)
f[p_, e_] := If[p == 2, 2^(e + 1) - 3, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)

a(n)=if(n<1, 0, (-1)^n*sumdiv(n,d,(-1)^d*d))

{a(n)=local(A,p,e); if(n<1, 0, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==2, 2^(e+1)-3, (p^(e+1)-1)/(p-1)))))}

Multiplicative with a(2^e) = 2^(e+1)-3 if e>0, a(p^e) = (p^(e+1)-1)/(p-1) if p>2.
G.f.: Sum_{k>0} -(-x)^k/(1+(-x)^k)^2 = Sum_{k>0} k*x^k/(1-(-x)^k).
Expansion of (1-(2/Pi)^2(2E(k)-K(k))K(k))/8 in powers of nome q where E(k) and K(k) are complete elliptic integrals and q=exp(-Pi*K(k')/K(k)). - Michael Somos, Jul 21 2006
Bisection: a(2*k-1) = A000203(2*k-1), a(2*k) = A146076(2*k) - A000593(2*k), k >= 1. See the Hardy reference where a(n) = sigma^*1(n). - _Wolfdieter Lang, Jan 07 2017
From Peter Bala, Dec 11 2020: (Start)
a(n) = Sum_{d | n, d != 2 (mod 4)} d.
O.g.f.: Sum_{k >= 1, k != 2 (mod 4)} k*x^k/(1 - x^k). Cf. A284362.
Define a(n) = 0 for n < 1. Then a(n) = e(n) + a(n-1) + a(n-3) - a(n-6) - a(n-10) + + - -, where [1, 3, 6, 10, ...] is the sequence of triangular numbers A000217, and e(n) = (-1)^(n+1)*n if n is a triangular number; otherwise e(n) = 0. Examples of this recurrence are given below. (End)
Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s) * zeta(s-1) * (1+2^(3-3*s)) / (1+2^(1-s)). - Werner Schulte, Jan 23 2021
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 16. - Vaclav Kotesovec, Aug 20 2021

Name corrected by Wolfdieter Lang, Jan 07 2017

A073007 Decimal expansion of Varga's constant.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

Equals the reciprocal of the one-ninth constant A072558.

Named after the American mathematician Richard Steven Varga (1928-2022). - Amiram Eldar, Jun 22 2021

			9.28902549192081891875544943595174506...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 4.5, p. 260.
R. S. Varga, Scientific Computation on Mathematical Problems and Conjectures, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 60, Philadelphia, PA: SIAM, 1990. See Chapter 2, pp. 23-38.

G. C. Greubel, Table of n, a(n) for n = 1..10000
A. J. Carpenter, A. Ruttan and R. S. Varga, Extended numerical computations on the "1/9" conjecture in rational approximation theory, in: P. R. Graves-Morris, E. B. Saff and R. S. Varga (eds.), Rational approximation and interpolation, Springer, Berlin, Heidelberg, 1984, pp. 383-411; alternative link.
Alphonse P. Magnus and Jean Meinguet, The elliptic functions and integrals of the '1/9' problem, Numerical Algorithms, Vol. 24, No. 1 (2000), pp. 117-139; alternative link.
Simon Plouffe, One-ninth constant.
Eric Weisstein's World of Mathematics, One-Ninth Constant.

Cf. A072558.

nmax=250; c = k /. FindRoot[EllipticK[k^2] == 2*EllipticE[k^2], {k, 9/10}, WorkingPrecision -> nmax]; Take[RealDigits[1/N[Exp[-Pi*(EllipticK[1 - c^2]/EllipticK[c^2])], nmax]][[1]], 200] (* G. C. Greubel, Mar 10 2018 *)
RealDigits[v /. FindRoot[4 EllipticE[InverseEllipticNomeQ[1/v]] == Pi EllipticTheta[3, 0, 1/v]^2, {v, 9, 9, 10}, WorkingPrecision -> 101]][[1]] (* Jan Mangaldan, Jun 25 2020 *)

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)

A143336 Expansion of K(k) * (2 * E(k) - K(k)) / (Pi/2)^2 in powers of q where E(k), K(k) are complete elliptic integrals and q = exp(-Pi * K(k') / K(k)).

Original entry on oeis.org

1, -8, -8, -32, -40, -48, -32, -64, -104, -104, -48, -96, -160, -112, -64, -192, -232, -144, -104, -160, -240, -256, -96, -192, -416, -248, -112, -320, -320, -240, -192, -256, -488, -384, -144, -384, -520, -304, -160, -448, -624, -336, -256, -352, -480, -624, -192, -384, -928, -456, -248, -576, -560, -432

Offset: 0

Michael Somos, Aug 09 2008

sign

			G.f. = 1 - 8*q - 8*q^2 - 32*q^3 - 40*q^4 - 48*q^5 - 32*q^6 - 64*q^7 - 104*q^8 + ...

Cf. A113184, A122858.

a[ n_] := If[ n < 1, Boole[n == 0], -(-1)^n 8 Sum[(-1)^d d, {d, Divisors @ n}]]; (* Michael Somos, Apr 07 2015 *)
a[ n_] := SeriesCoefficient[ With[{m = InverseEllipticNomeQ[ q]}, EllipticK[ m] (2 EllipticE[ m] - EllipticK[ m]) (2/Pi)^2], {q, 0, n}]; (* Michael Somos, Apr 07 2015 *)

{a(n) = if( n<1, n==0, -(-1)^n * 8 * sumdiv(n, d, (-1)^d * d))};

The generating function equals 0 when 2 * E(k) = K(k) at q = 0.1076539192... (A072558) the "One-Ninth" constant.
Expansion of (P(q) - 2 * P(q^2) + 4 * P(q^4)) / 3 in powers of q where P() is a Ramanujan Lambert series.
G.f.: 1 - 8 * Sum_{k>0} k * x^k / (1 - (-x)^k) = 1 + 8 * Sum_{k>0} (-x)^k / (1 + (-x)^k)^2.
a(n) = (-1)^n * A122858(n). a(n) = -8 * A113184(n) unless n=0.

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

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A113184 Absolute difference between sum of odd divisors of n and sum of even divisors of n.

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A073007 Decimal expansion of Varga's constant.

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

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A143336 Expansion of K(k) * (2 * E(k) - K(k)) / (Pi/2)^2 in powers of q where E(k), K(k) are complete elliptic integrals and q = exp(-Pi * K(k') / K(k)).

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