A093178 If n is even then 1, otherwise n.
1, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, 1, 21, 1, 23, 1, 25, 1, 27, 1, 29, 1, 31, 1, 33, 1, 35, 1, 37, 1, 39, 1, 41, 1, 43, 1, 45, 1, 47, 1, 49, 1, 51, 1, 53, 1, 55, 1, 57, 1, 59, 1, 61, 1, 63, 1, 65, 1, 67, 1, 69, 1, 71, 1, 73, 1, 75, 1, 77, 1, 79, 1, 81, 1, 83, 1, 85
Offset: 0
Examples
1.557407724654902230506974807... = 1 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + ...)))) G.f. = 1 + x + x^2 + 3*x^3 + x^4 + 5*x^5 + x^6 + 7*x^7 + x^8 + 9*x^9 + x^10 + ...
Links
- Harry J. Smith, Table of n, a(n) for n = 0..20000
- D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
- Simon Plouffe, A Search for a mathematical expression for mass ratios using a large database. page 3.
- G. Xiao, Contfrac
- Index entries for continued fractions for constants
- Index entries for two-way infinite sequences
- Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
Crossrefs
Programs
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Maple
A093178:=n->(n+1+(1-n)*(-1)^n)/2; seq(A093178(k), k=0..100); # Wesley Ivan Hurt, Oct 19 2013
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Mathematica
Join[{1},Riffle[Range[1,85,2],1]] (* or *) Array[If[EvenQ[#],1,#]&,87,0] (* Harvey P. Dale, Nov 23 2011 *) -
PARI
{a(n) = if( n%2, n, 1)};
Formula
G.f.: (1+x-x^2+x^3)/(1-x^2)^2.
a(n) = (-1)^n * a(-n) for all n in Z.
a(n) = (1/2) * [ 1 + n + (1-n)*(-1)^n ]. - Ralf Stephan, Dec 02 2004
a(n) = n^n mod (n+1) for n > 0. - Amarnath Murthy, Apr 18 2004
Satisfies a(0) = 1, a(n+1) = a(n) + n if a(n) < n else a(n+1) = a(n)/n. - Amarnath Murthy, Oct 29 2002
a(n) = ((n+1)+(1-n)(-1)^n)/2 and have e.g.f. (1+x)cosh(x). - Paul Barry, Apr 09 2003
a(n) = binomial(n, 2*floor(n/2)). - Paul Barry, Dec 28 2006
Starting (1, 1, 3, 1, 5, 1, 7, ...) = A133080^(-1) * [1,2,3,...]. - Gary W. Adamson, Sep 08 2007
a(n) = denom(b(n+2)/b(n+1)) with b(n) = product((2*n-3-2*k), k=0..floor(n/2-1)). - Johannes W. Meijer, Jun 18 2009
a(n) = 2*floor(n/2) - n*(n-1 mod 2) + 1. - Wesley Ivan Hurt, Oct 19 2013
a(n) = n^(n mod 2). - Wesley Ivan Hurt, Apr 16 2014
Comments