A093178 - OEIS

%I #80 Dec 06 2024 07:02:29

%S 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,

%T 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,

%U 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

%N If n is even then 1, otherwise n.

%C Continued fraction expansion for tan(1).

%C 1 followed by run lengths of A062557 = 2n-1 1's followed by a 2. - _Jeremy Gardiner_, Aug 12 2012

%C Greatest common divisor of n and (n+1) mod 2. - _Bruno Berselli_, Mar 07 2017

%H Harry J. Smith, <a href="/A093178/b093178.txt">Table of n, a(n) for n = 0..20000</a>

%H D. H. Lehmer, <a href="/A016825/a016825.pdf">Continued fractions containing arithmetic progressions</a>, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]

%H Simon Plouffe, <a href="https://vixra.org/pdf/1409.0099v1.pdf">A Search for a mathematical expression for mass ratios using a large database</a>. page 3.

%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>

%H <a href="/index/Con#confC">Index entries for continued fractions for constants</a>

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).

%F G.f.: (1+x-x^2+x^3)/(1-x^2)^2.

%F a(n) = (-1)^n * a(-n) for all n in Z.

%F a(n) = (1/2) * [ 1 + n + (1-n)*(-1)^n ]. - _Ralf Stephan_, Dec 02 2004

%F a(n) = n^n mod (n+1) for n > 0. - _Amarnath Murthy_, Apr 18 2004

%F 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

%F a(n) = ((n+1)+(1-n)(-1)^n)/2 and have e.g.f. (1+x)cosh(x). - _Paul Barry_, Apr 09 2003

%F a(n) = binomial(n, 2*floor(n/2)). - _Paul Barry_, Dec 28 2006

%F Starting (1, 1, 3, 1, 5, 1, 7, ...) = A133080^(-1) * [1,2,3,...]. - _Gary W. Adamson_, Sep 08 2007

%F 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

%F a(n) = 2*floor(n/2) - n*(n-1 mod 2) + 1. - _Wesley Ivan Hurt_, Oct 19 2013

%F a(n) = n^(n mod 2). - _Wesley Ivan Hurt_, Apr 16 2014

%e 1.557407724654902230506974807... = 1 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + ...))))

%e 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 + ...

%p A093178:=n->(n+1+(1-n)*(-1)^n)/2; seq(A093178(k), k=0..100); # _Wesley Ivan Hurt_, Oct 19 2013

%t Join[{1},Riffle[Range[1,85,2],1]] (* or *) Array[If[EvenQ[#],1,#]&,87,0] (* _Harvey P. Dale_, Nov 23 2011 *)

%o (PARI) {a(n) = if( n%2, n, 1)};

%Y Equals |A009001(n)|.

%Y Cf. A133080, A049471 (decimal expansion), A009001, A161738, A062557, A124625.

%K nonn,easy

%O 0,4

%A _Michael Somos_, Mar 27 2004