This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A100261 #26 Jun 22 2025 03:09:44 %S A100261 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 A100261 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 A100261 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 A100261 Continued fraction expansion of cot(1-Pi/4). %D A100261 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 A100261 Antti Karttunen, <a href="/A100261/b100261.txt">Table of n, a(n) for n = 1..16384</a> %H A100261 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 A100261 <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 A100261 This number is also -Im[ (e^(2i)+i)/(e^(2i)-i) ]. %F A100261 Periodic part is ...2, 3k+2, 1, 12k+16, 1, 3k+4, ... (k=0..oo). %F A100261 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 %F A100261 cot(1 - Pi/4) = (sin(1) + cos(1))/((sin(1) - cos(1))) = A143623/|A143624|. - _Peter Bala_, Jun 17 2025 %e A100261 4.588037824983899981397906503733748769677138839382189177607356840... %t A100261 ContinuedFraction[ -Im[(E^(2I) + I)/(E^(2I) - I)], 80] (* _Robert G. Wilson v_, Nov 20 2004 *) %t A100261 ContinuedFraction[Cot[1-Pi/4],100] (* _Harvey P. Dale_, Feb 26 2025 *) %o A100261 (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 A100261 Cf. A005131. %K A100261 nonn,cofr,easy %O A100261 1,1 %A A100261 _Ralf Stephan_, Nov 18 2004