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A133265 Diagonal of the A135356 triangle.

%I A133265 #25 Sep 08 2022 08:45:31
%S A133265 2,2,2,4,2,6,2,8,2,10,2,12,2,14,2,16,2,18,2,20,2,22,2,24,2,26,2,28,2,
%T A133265 30,2,32,2,34,2,36,2,38,2,40,2,42,2,44,2,46,2,48,2,50,2,52,2,54,2,56,
%U A133265 2,58,2,60,2,62,2,64,2,66,2,68,2,70,2,72,2,74,2,76,2,78,2,80
%N A133265 Diagonal of the A135356 triangle.
%C A133265 Regular continued fraction expansion of 2*sin(1/2)/( cos(1/2) - sin(1/2) ) = 2.40822 34423 35827 84841 ... = 2 + 1/(2 + 1/(2 + 1/(4 + 1/(2 + 1/(6 + 1/(2 + 1/(8 + 1/(2 + ... )))))))). Cf. A019425. - _Peter Bala_, Feb 15 2015
%H A133265 Vincenzo Librandi, <a href="/A133265/b133265.txt">Table of n, a(n) for n = 0..1000</a>
%H A133265 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A133265 2*(A057979 without 1, 0, first two terms).
%F A133265 a(n) = (n+3+(n-1)*(-1)^(n+1))/2. - _Vincenzo Librandi_, Aug 30 2011
%F A133265 a(n) = (2*n + 5) mod (n*(3 + (-1)^n) - (-1)^n + 7)/2. - _Lechoslaw Ratajczak_, Nov 17 2016
%F A133265 From _Colin Barker_, Nov 17 2016: (Start)
%F A133265 a(n) = 2*a(n-2) - a(n-4) for n>3.
%F A133265 G.f.: 2*(1 + x - x^2) / ((1 - x)^2 * (1 + x)^2).
%F A133265 (End)
%p A133265 A133265 := n -> (n+2+(n-2)*(-1)^n)/2: # _Peter Luschny_, Aug 30 2011
%t A133265 Table[(n + 3 + (n - 1) (-1)^(n + 1))/2, {n, 0, 79}] (* or *)
%t A133265 Table[Mod[(2 n + 5), (n (3 + (-1)^n) - (-1)^n + 7)/2], {n, 0, 79}] (* or *)
%t A133265 CoefficientList[Series[2 (1 + x - x^2)/((1 - x)^2*(1 + x)^2), {x, 0, 79}], x] (* _Michael De Vlieger_, Nov 18 2016 *)
%o A133265 (Magma) [(n+3+(n-1)*(-1)^(n+1))/2: n in [0..80]]; // _Vincenzo Librandi_, Aug 30 2011
%o A133265 (PARI) Vec(2*(1 + x - x^2) / ((1 - x)^2 * (1 + x)^2) + O(x^100)) \\ _Colin Barker_, Nov 17 2016
%Y A133265 Cf. A019425.
%K A133265 nonn,easy
%O A133265 0,1
%A A133265 _Paul Curtz_, Dec 20 2007