%I #20 Mar 15 2024 10:11:59
%S 4,1,1,2,2,1,16,1,4,2,5,1,28,1,7,2,8,1,40,1,10,2,11,1,52,1,13,2,14,1,
%T 64,1,16,2,17,1,76,1,19,2,20,1,88,1,22,2,23,1,100,1,25,2,26,1,112,1,
%U 28,2,29,1,124,1,31,2,32,1,136,1,34,2,35,1,148,1,37,2,38,1,160,1,40
%N Continued fraction expansion of cot(1-Pi/4).
%D Lipshitz, Leonard, and A. van der Poorten. "Rational functions, diagonals, automata and arithmetic." In Number Theory, Richard A. Mollin, ed., Walter de Gruyter, Berlin (1990): 339-358.
%H Antti Karttunen, <a href="/A100261/b100261.txt">Table of n, a(n) for n = 1..16384</a>
%H A. J. Van der Poorten, <a href="https://web.williams.edu/Mathematics/sjmiller/public_html/book/papers/vdp/Poorten_CFExpansionsValuesOfExp.pdf">Continued fraction expansions of values of the exponential function and related fun with continued fractions</a>, Nieuw Archief voor Wiskunde, Vol. 14 (1996), pp. 221-230.
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,2,0,0,0,0,0,-1).
%F This number is also -Im[ (e^(2i)+i)/(e^(2i)-i) ].
%F Periodic part is ...2, 3k+2, 1, 12k+16, 1, 3k+4, ... (k=0..oo).
%F G.f.: -x*(x^11-x^10+2*x^9-2*x^8+x^7-8*x^6-x^5-2*x^4-2*x^3-x^2-x-4) / ((x-1)^2*(x+1)^2*(x^2-x+1)^2*(x^2+x+1)^2). - _Colin Barker_, Jul 15 2013
%e 4.588037824983899981397906503733748769677138839382189177607356840...
%t ContinuedFraction[ -Im[(E^(2I) + I)/(E^(2I) - I)], 80] (* _Robert G. Wilson v_, Nov 20 2004 *)
%o (PARI) A100261(n) = if(1==n,4,if(n<4,1, n=n-4; my(k=n\6); if(!(n%6), 2, if(1==(n%6), 3*k + 2, if(3==(n%6), 12*k + 16, if(5==(n%6), 3*k + 4, 1)))))); \\ _Antti Karttunen_, Feb 15 2023
%Y Cf. A005131.
%K nonn,cofr,easy
%O 1,1
%A _Ralf Stephan_, Nov 18 2004