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A103710
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Decimal expansion of the ratio of the latus rectum arc of any parabola to its semi latus rectum.
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5
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2, 2, 9, 5, 5, 8, 7, 1, 4, 9, 3, 9, 2, 6, 3, 8, 0, 7, 4, 0, 3, 4, 2, 9, 8, 0, 4, 9, 1, 8, 9, 4, 9, 0, 3, 8, 7, 5, 9, 7, 8, 3, 2, 2, 0, 3, 6, 3, 8, 5, 8, 3, 4, 8, 3, 9, 2, 9, 9, 7, 5, 3, 4, 6, 6, 4, 4, 1, 0, 9, 6, 6, 2, 6, 8, 4, 1, 3, 3, 1, 2, 6, 6, 8, 4, 0, 9, 4, 4, 2, 6, 2, 3, 7, 8, 9, 7, 6, 1, 5, 5, 9, 1, 7, 5
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| All parabolas are similar (Ogilvy, 1969). Just as the ratio of a semicircle to its radius is always pi, the ratio of the latus rectum arc of any parabola to its semi latus rectum is sqrt(2) + ln(1 + sqrt(2)).
The Universal Parabolic Constant, equal to the ratio of the latus rectum arc of any parabola to its focal parameter. Like pi, it is transcendental.
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REFERENCES
| C. E. Love, Differential and Integral Calculus, 4th ed., Macmillan, 1950, pp. 286-288.
C. S. Ogilvy, Excursions in Geometry, Oxford Univ. Press, 1969, p. 84.
S. Reese, A universal parabolic constant, 2004, preprint.
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LINKS
| S. R. Finch, Mathematical Constants, addenda, section 8.1
S. Reese, Pohle Colloquium Video Lecture: The universal parabolic constant, February 2, 2005
Eric Weisstein's World of Mathematics, Universal Parabolic Constant
Eric Weisstein et al., Universal Parabolic Constant
Wikipedia, Universal parabolic constant
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FORMULA
| sqrt(2) + ln(1 + sqrt(2)).
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EXAMPLE
| 2.29558714939263807403429804918949038759783220363858348392997534664...
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MATHEMATICA
| RealDigits[ Sqrt[2] + Log[1 + Sqrt[2]], 10, 111][[1]] (from Robert G. Wilson v Feb 14 2005)
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CROSSREFS
| Cf. A002193 + A091648.
See also A103711, A103712.
Sequence in context: A157216 A020776 A021002 * A178236 A093589 A073315
Adjacent sequences: A103707 A103708 A103709 * A103711 A103712 A103713
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KEYWORD
| cons,easy,nonn
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AUTHOR
| Sylvester Reese and Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Feb 13 2005
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