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A123167
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Continued fraction for c=sqrt(2)*(exp(sqrt(2))+1)/(exp(sqrt(2))-1). a(2n-1)=8n-6, a(2n)=4n-1.
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2
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2, 3, 10, 7, 18, 11, 26, 15, 34, 19, 42, 23, 50, 27, 58, 31, 66, 35, 74, 39, 82, 43, 90, 47, 98, 51, 106, 55, 114, 59, 122, 63, 130, 67, 138, 71, 146, 75, 154, 79, 162, 83, 170, 87, 178, 91, 186, 95, 194, 99, 202, 103, 210, 107, 218, 111, 226, 115, 234, 119, 242, 123
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| This continued fraction shows exp(sqrt(2)) is irrationnal.
If a(0)=-1 and offset 0: a(6*n)-a(6*n+1)+a(6*n+2)=0 , a(6*n +3)-4*a(6*n+4)+a(6*n+5)=0.
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REFERENCES
| J. Borwein and D. Bailey, Mathematics by experiment, plausible reasoning in the 21st Century, A. K. Peters, p. 77
J. Borwein and K. Devlin, The computer as crucible: an introduction to experimental mathematics, A. K. Peters 2009, p. 91.
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FORMULA
| a(n) = - A123168(2 - n) unless n = 1. - Michael Somos, Feb 24 2012
Empirical G.f. and recurrence: x*(2+3*x+6*x^2+x^3)/(1-2*x^2+x^4). a(n)=2*a(n-2)-a(n-4).[Colin Barker, Feb 08 2012]
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EXAMPLE
| c = 2.3227261394604270...
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PROG
| (PARI) {a(n) = (2*n - 1) * 2^(n%2)} /* Michael Somos, Feb 04 2012 */
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CROSSREFS
| Cf. A123168.
Sequence in context: A078730 A163767 A128531 * A141670 A193729 A074068
Adjacent sequences: A123164 A123165 A123166 * A123168 A123169 A123170
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KEYWORD
| nonn,cofr,changed
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AUTHOR
| Benoit Cloitre (abmt(AT)orange.fr), Oct 02 2006
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