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A240969
Decimal expansion of the breadth of the "caliper", the broadest worm of unit length.
3
4, 3, 8, 9, 2, 5, 3, 6, 9, 2, 5, 9, 4, 6, 6, 4, 5, 6, 7, 4, 0, 8, 8, 5, 2, 6, 1, 1, 5, 8, 5, 2, 3, 7, 7, 4, 2, 1, 9, 1, 4, 9, 3, 8, 6, 5, 1, 4, 3, 8, 8, 7, 2, 6, 8, 3, 0, 1, 0, 7, 5, 9, 7, 5, 2, 9, 2, 6, 0, 4, 4, 2, 0, 4, 9, 2, 6, 6, 8, 7, 2, 4, 6, 0, 3, 3, 0, 0, 4, 1, 3, 7, 5, 7, 9, 1, 4, 9, 2, 2
OFFSET
0,1
COMMENTS
A caliper consists of 2 circular arcs with 4 tangent segments, specifically configured (see link to Figure 8.3 from the book by Steven Finch).
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.4 Moser's Worm Constant, p. 493.
FORMULA
See trig. formulas in Mathematica code.
Sec(phi), an algebraic number, is the positive root of 3x^6 + 36x^4 + 16x^2 - 64.
EXAMPLE
phi = 0.29004634452825946320905124629823276955932638591519522257237...
psi = 0.480931237564380337681715512959999015584157793267187574483...
beta = 0.43892536925946645674088526115852377421914938651438872683...
MATHEMATICA
phi = ArcSin[1/6 + (4/3)*Sin[(1/3 )*ArcSin[17/64]]];
psi = ArcTan[(1/2)*Sec[phi]];
beta = (1/2)*(Pi/2 - phi - 2*psi + Tan[phi] + Tan[psi])^(-1);
RealDigits[beta, 10, 100] // First
CROSSREFS
Cf. A227472.
Sequence in context: A004125 A137924 A171527 * A198576 A368551 A175047
KEYWORD
nonn,cons
AUTHOR
STATUS
approved