A244662 Decimal expansion of 'C' (as designated by D. Shanks), a constant appearing in the second order term of the asymptotic expansion of the number of non-hypotenuse numbers not exceeding a given bound.

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

Offset

0,1

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 2.3 Landau-Ramanujan Constant, p. 101.

Links

Table of n, a(n) for n=0..99.

Daniel Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation, Vol. 18 (1964), pp. 75-86.

Daniel Shanks, Non-hypotenuse Numbers, Fib. Quart., 13:4 (1975), pp. 319-321.

Eric Weisstein's MathWorld, Lemniscate Constant

Formula

C = c + 1/2*log((Pi/L)^2*exp(gamma)/2), where c is A227158 and L the Lemniscate constant A062539.

Example

0.70475345170594788412255819759189881852...

Mathematica

digits = 100; m0 = 5; dm = 5; beta[x_] := 1/4^x*(Zeta[x, 1/4] - Zeta[x, 3/4]); L = Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2; Clear[f]; f[m_] := f[m] = 1/2*(1 - Log[Pi*E^EulerGamma/(2*L)]) - 1/4*NSum[Zeta'[2^k]/Zeta[2^k] - beta'[2^k]/beta[2^k] + Log[2]/(2^(2^k) - 1), {k, 1, m}, WorkingPrecision -> digits + 10]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits] != RealDigits[f[m - dm], 10, digits], m = m + dm]; c = A227158 = f[m]; c + 1/2 Log[(Pi/L)^2*Exp[EulerGamma]/2] // RealDigits[#, 10, digits] & // First

Crossrefs

Cf. A009003, A004144, A062539, A227158, A244659 (first order term).

Sequence in context: A374776 A177156 A016582 * A060708 A021997 A099737

Adjacent sequences: A244659 A244660 A244661 * A244663 A244664 A244665

Keyword

nonn,cons

Author

Jean-François Alcover, Jul 04 2014

Status

approved