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A103711
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Decimal expansion of the ratio of the latus rectum arc of any parabola to its latus rectum.
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3
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1, 1, 4, 7, 7, 9, 3, 5, 7, 4, 6, 9, 6, 3, 1, 9, 0, 3, 7, 0, 1, 7, 1, 4, 9, 0, 2, 4, 5, 9, 4, 7, 4, 5, 1, 9, 3, 7, 9, 8, 9, 1, 6, 1, 0, 1, 8, 1, 9, 2, 9, 1, 7, 4, 1, 9, 6, 4, 9, 8, 7, 6, 7, 3, 3, 2, 2, 0, 5, 4, 8, 3, 1, 3, 4, 2, 0, 6, 6, 5, 6, 3, 3, 4, 2, 0, 4, 7, 2, 1, 3, 1, 1, 8, 9, 4, 8, 8, 0, 7, 7, 9, 5, 8, 7
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| All parabolas are similar (Ogilvy, 1969). Just as the ratio of a semicircle to its diameter is always pi/2, the ratio of the latus rectum arc of any parabola to its latus rectum is (sqrt(2) + ln(1 + sqrt(2)))/2.
Let c = this constant and a = e - exp((c+Pi)/2 - ln(Pi)), then a = .0000999540234051652627... and c - 10*(-ln(exp(a) - a - 1) - 19) = .000650078964115564700067717... - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Feb 21 2005
Half the Universal Parabolic Constant A103710 (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)))/2.
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EXAMPLE
| 1.14779357469631903701714902459474519379891610181929174196498767332...
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MATHEMATICA
| RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/2, 10, 111][[1]] (from Robert G. Wilson v Feb 14 2005)
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CROSSREFS
| Equal to (A103710)/2 = (A002193 + A091648)/2 = 3*(A103712).
Sequence in context: A011222 A157298 A070326 * A199435 A159919 A131432
Adjacent sequences: A103708 A103709 A103710 * A103712 A103713 A103714
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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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