A166639 - OEIS

%I #39 Feb 25 2023 05:24:27

%S 7,5,19,17,31,29,43,41,55,53,67,65,79,77,91,89,103,101,115,113,127,

%T 125,139,137,151,149,163,161,175,173,187,185,199,197,211,209,223,221,

%U 235,233,247,245,259,257,271,269,283,281,295,293,307,305,319,317,331,329

%N a(n) = 6*n + 3 + 4*(-1)^n.

%C Prime pairs in this sequence of form p, p+14 (5,19 - 17,31 - 29,43 - 53,67 - 89,103 - 113,127 - 137,151 - 149,163 - 197,211 - 257, 271 - 269,283 and so on).

%C The numerators in the continued fraction of 2^(7/12) = 1 + 7/(12 + 5/(2 + 19/(36 + 17/(2 + 31/(60 + 29/(2 + 43/(84 + 41/(2 + ...)))))))). - _Philippe Deléham_, Nov 02 2013

%H Vincenzo Librandi, <a href="/A166639/b166639.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F G.f.: (7-2*x+7*x^2) / ( (1+x)*(x-1)^2 ). - _R. J. Mathar_, Feb 07 2011

%F From _G. C. Greubel_, May 20 2016: (Start)

%F E.g.f.: (6*x + 7)*cosh(x) + (6*x - 1)*sinh(x).

%F a(n) = a(n-1) + a(n-2) - a(n-3). (End)

%F Sum_{n>=0} (-1)^(n+1)/a(n) = (1/6 - 1/(4*sqrt(3)))*Pi. - _Amiram Eldar_, Feb 25 2023

%t Table[(6 n + 3 + 4*(-1)^n), {n, 0, 80}] (* _Vincenzo Librandi_, Sep 15 2013 *)

%t LinearRecurrence[{1,1,-1},{7,5,19},50] (* _G. C. Greubel_, May 20 2016 *)

%o (Magma) [6*n+3+4*(-1)^n: n in [0..60]]; // _Vincenzo Librandi_, Sep 15 2013

%K nonn,easy

%O 0,1

%A _Vincenzo Librandi_, Oct 18 2009

%E Corrected sign on the recommendation of _Zak Seidov_

%E Changed offset from 1 to 0 and added a(0) by _Vincenzo Librandi_, Sep 15 2013

%E Adapted G.f. to new offset by _Vincenzo Librandi_, Sep 15 2013