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 A175886 #50 Sep 03 2025 09:15:17
%S A175886 1,12,14,25,27,38,40,51,53,64,66,77,79,90,92,103,105,116,118,129,131,
%T A175886 142,144,155,157,168,170,181,183,194,196,207,209,220,222,233,235,246,
%U A175886 248,259,261,272,274,285,287,298,300,311,313,324,326,337,339,350
%N A175886 Numbers that are congruent to {1, 12} mod 13.
%C A175886 Cf. property described by _Gary Detlefs_ in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1 == 0 (mod h); in this case, a(n)^2-1 == 0 (mod 13).
%H A175886 Bruno Berselli, <a href="/A175886/b175886.txt">Table of n, a(n) for n = 1..10000</a>
%H A175886 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A175886 G.f.: x*(1+11*x+x^2)/((1+x)*(1-x)^2).
%F A175886 a(n) = (26*n+9*(-1)^n-13)/4.
%F A175886 a(n) = -a(-n+1) = a(n-1)+a(n-2)-a(n-3).
%F A175886 a(n) = a(n-2)+13.
%F A175886 a(n) = 13*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n>1.
%F A175886 Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/13)*cot(Pi/13). - _Amiram Eldar_, Dec 04 2021
%F A175886 E.g.f.: 1 + ((26*x - 13)*exp(x) + 9*exp(-x))/4. - _David Lovler_, Sep 04 2022
%F A175886 From _Amiram Eldar_, Nov 25 2024: (Start)
%F A175886 Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/13).
%F A175886 Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/13)*cosec(Pi/13). (End)
%t A175886 Select[Range[1, 350], MemberQ[{1, 12}, Mod[#, 13]]&] (* _Bruno Berselli_, Feb 29 2012 *)
%t A175886 CoefficientList[Series[(1 + 11 x + x^2) / ((1 + x) (1 - x)^2), {x, 0, 55}], x] (* _Vincenzo Librandi_, Aug 19 2013 *)
%t A175886 LinearRecurrence[{1,1,-1},{1,12,14},60] (* _Harvey P. Dale_, Oct 23 2015 *)
%o A175886 (Haskell)
%o A175886 a175886 n = a175886_list !! (n-1)
%o A175886 a175886_list = 1 : 12 : map (+ 13) a175886_list
%o A175886 -- _Reinhard Zumkeller_, Jan 07 2012
%o A175886 (Magma) [n: n in [1..350] | n mod 13 in [1, 12]]; // _Bruno Berselli_, Feb 29 2012
%o A175886 (Magma) [(26*n+9*(-1)^n-13)/4: n in [1..55]]; // _Vincenzo Librandi_, Aug 19 2013
%o A175886 (PARI) a(n)=(26*n+9*(-1)^n-13)/4 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A175886 Cf. A000217, A091998, A113801, A005408, A047209, A007310, A047336, A047522, A056020, A090771, A175885, A175887.
%Y A175886 Cf. A195045 (partial sums).
%K A175886 nonn,easy,changed
%O A175886 1,2
%A A175886 _Bruno Berselli_, Oct 08 2010 - Nov 17 2010