|
| |
|
|
A085835
|
|
Decimal expansion of Grossman's constant.
|
|
1
|
|
|
|
7, 3, 7, 3, 3, 8, 3, 0, 3, 3, 6, 9, 2, 8, 4, 9, 6
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
COMMENTS
|
This is the unique x such that the sequence s_0=1, s_1=x and s_n = s_{n-2}/(1+s_{n-1}) for n >= 2 converges.
|
|
|
REFERENCES
|
S. R. Finch, "Grossman's Constant", Section 6.4 in Mathematical Constants, Cambridge University Press, pp. 429-430, 2003.
Grossman, J. W. "Problem 86-2." Math. Intel. 8, 31, 1986.
Janssen, A. J. E. M. and Tjaden, D. L. A. Solution to Problem 86-2. Math. Intel. 9, 40-43, 1987.
|
|
|
LINKS
|
Table of n, a(n) for n=0..16.
Eric Weisstein's World of Mathematics, Grossman's Constant
|
|
|
EXAMPLE
|
0.737338...
|
|
|
MATHEMATICA
|
(* Discrete Fourier transform is used to evaluate smoothness of sequence s(n, x) *) x0 = SetPrecision[0.7373, precis = 32]; digits = 17; m0 =(* initial number of terms = *)1000; defineSeq = (Clear[s, f]; s[0, _] = 1; s[1, x_] := s[1, x] = x; s[n_, x_] := s[n, x] = s[n-2, x]/(1 + s[n-1, x]); f[x_, m_] := f[x, m] = Min[Abs[Fourier[Table[s[n, x], {n, 1, m}]]]]); rd[-1] = RealDigits[x0] // First; Do[defineSeq; j = If[k == 0, 4, k+6]; dx = 10^-j; x1 = x0-2*dx; x2 = x0-dx; x3 = x0+dx; x4 = x0+2*dx; m = 2^k*m0; x0 = SetPrecision[(x2*(x4-x3)*f[x1, m] + x1*(x4-x3)* f[x2, m] + (x1-x2)*(x4*f[x3, m] - x3*f[x4, m])) / ((x4-x3)*f[x1, m] + (x3-x4)*f[x2, m] + (x1-x2)*(f[x3, m] - f[x4, m])), precis]; rd[k] = RealDigits[x0] // First; lg = Min[digits, rd[k-1] // Length, rd[k] // Length]; Print["k = ", k, " x0 = ", x0, " number of terms examined = ", m]; A085835 = Take[rd[k], lg]; If[ A085835 == Take[rd[k-1], lg], Break[]], {k, 0, digits}]; A085835 (* Jean-François Alcover, Nov 29 2012 *)
|
|
|
CROSSREFS
|
Sequence in context: A173182 A021856 A096715 * A153624 A160578 A166201
Adjacent sequences: A085832 A085833 A085834 * A085836 A085837 A085838
|
|
|
KEYWORD
|
nonn,cons,more
|
|
|
AUTHOR
|
Eric W. Weisstein, Jul 05, 2003
|
|
|
EXTENSIONS
|
Extended from 14 to 17 digits by Jean-François Alcover, Nov 22 2012
|
|
|
STATUS
|
approved
|
| |
|
|
