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A123167
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Continued fraction for c=sqrt(2)*(exp(sqrt(2))+1)/(exp(sqrt(2))-1). a(2*n-1) = 8*n-6, a(2*n) = 4*n-1.
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4
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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
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OFFSET
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1,1
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COMMENTS
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This continued fraction shows exp(sqrt(2)) is irrational.
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
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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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LINKS
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FORMULA
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Empirical g.f.: x*(2+3*x+6*x^2+x^3)/(1-2*x^2+x^4).
Empirical a(n) = 2*a(n-2) - a(n-4). (End)
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EXAMPLE
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c = 2.3227261394604270...
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MAPLE
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if type(n, 'even') then
2*n-1 ;
else
4*n-2 ;
end if;
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MATHEMATICA
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a[ n_] := (2 n - 1) 2^Mod[n, 2]; (* Michael Somos, Apr 25 2015 *)
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PROG
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(PARI) {a(n) = (2*n - 1) * 2^(n%2)}; \\ Michael Somos, Feb 04 2012
(Magma) [(2*n-1)*2^(n mod 2): n in [1..50]]; // G. C. Greubel, Jan 27 2018
(GAP) a := [2, 3, 10, 7];; for n in [5..10^3] do a[n] := 2*a[n-2] - a[n-4]; od; a; # Muniru A Asiru, Jan 28 2018
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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STATUS
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approved
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