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A123167
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.
4
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
OFFSET
1,1
COMMENTS
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.
Conjecture: Numerator of 4/n - 2/n^2. - Wesley Ivan Hurt, Jul 11 2016
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.
FORMULA
a(n) = - A123168(2 - n) for all n in Z unless n = 1. - Michael Somos, Feb 24 2012
From Colin Barker, Feb 08 2012: (Start)
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)
EXAMPLE
c = 2.3227261394604270...
MAPLE
A123167 := proc(n)
if type(n, 'even') then
2*n-1 ;
else
4*n-2 ;
end if;
end proc: # R. J. Mathar, Jul 25 2013
MATHEMATICA
a[ n_] := (2 n - 1) 2^Mod[n, 2]; (* Michael Somos, Apr 25 2015 *)
PROG
(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
CROSSREFS
Cf. A123168.
Sequence in context: A336091 A347835 A128531 * A333176 A338043 A141670
KEYWORD
nonn,cofr
AUTHOR
Benoit Cloitre, Oct 02 2006
STATUS
approved