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%I #45 Jan 01 2024 23:47:02
%S 0,2,6,10,14,18,22,26,30,34,38,42,46,50,54,58,62,66,70,74,78,82,86,90,
%T 94,98,102,106,110,114,118,122,126,130,134,138,142,146,150,154,158,
%U 162,166,170,174,178,182,186,190,194,198,202,206,210,214,218,222,226,230
%N a(n) = 2*A004273(n).
%C Equals A111284 from the 2nd term on. - _R. J. Mathar_, Jun 13 2008
%C Besides the first term, this sequence gives the denominators of the alternating series Pi/8 = 1/2 - 1/6 + 1/10 - 1/14 + 1/18 - 1/22 + .... - _Mohammad K. Azarian_, Oct 14 2011 [edited by _Jon E. Schoenfield_, Mar 07 2015]
%C Numbers that cannot be a side of a primitive Pythagorean triangle. - _Torlach Rush_, Nov 07 2019
%C Simple continued fraction expansion of tanh(1/2) = (e - 1)/(e + 1) = 1/(2 + 1/(6 + 1/(10 + 1/(14 + ...)))). - _Peter Bala_, Oct 01 2023
%D Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).
%H Vincenzo Librandi, <a href="/A130824/b130824.txt">Table of n, a(n) for n = 0..10000</a>
%H Mohammad K. Azarian, <a href="https://www.jstor.org/stable/24340810">Problem 1218</a>, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. <a href="https://www.jstor.org/stable/24337914">Solution</a> published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F From _Stefano Spezia_, Dec 09 2019: (Start)
%F G.f.: 2*x*(1+x)/(1-x)^2.
%F a(n) = 2*a(n-1) - a(n-2) for n > 0.
%F a(n) = 4*n - 1 - (-1)^(2^n). (End)
%F E.g.f: 2*(1 - (1-2*x)*exp(x)). - _G. C. Greubel_, Dec 30 2019
%p A130827 := proc(n) if n =0 then 0 ; else 4*n-2 ; fi ; end: seq(A130827(n),n=0..120) ; # _R. J. Mathar_, Oct 28 2007
%t 2 Join[{0}, Range[1, 200, 2]] (* _Michael De Vlieger_, Mar 07 2015 *)
%o (Magma) [4*n-2*Floor((n+2) mod (n+1)):n in [0..60]]; // _Vincenzo Librandi_, Sep 22 2011
%o (PARI) vector(61, n, if(n==1, 0, 4*(n-1) -2) ) \\ _G. C. Greubel_, Dec 30 2019
%o (Sage) [0]+[4*n-2 for n in (1..60)] # _G. C. Greubel_, Dec 30 2019
%o (GAP) Concatenation([0], List([1..60], n-> 4*n-2 )); # _G. C. Greubel_, Dec 30 2019
%Y Cf. A016825, A111284, A160327.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, Jul 17 2007
%E More terms from _R. J. Mathar_, Oct 28 2007
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