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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 (list; graph; refs; listen; history; text; internal format)
 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. LINKS G. C. Greubel, Table of n, a(n) for n = 1..10000 Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1). 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: A292239 A163767 A128531 * A333176 A141670 A278561 Adjacent sequences:  A123164 A123165 A123166 * A123168 A123169 A123170 KEYWORD nonn,cofr AUTHOR Benoit Cloitre, Oct 02 2006 STATUS approved

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Last modified May 27 18:56 EDT 2020. Contains 334664 sequences. (Running on oeis4.)