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 A177065 #33 Oct 25 2024 14:36:57
%S A177065 15,143,399,783,1295,1935,2703,3599,4623,5775,7055,8463,9999,11663,
%T A177065 13455,15375,17423,19599,21903,24335,26895,29583,32399,35343,38415,
%U A177065 41615,44943,48399,51983,55695,59535,63503,67599,71823,76175,80655,85263,89999,94863,99855
%N A177065 a(n) = (8*n+3)*(8*n+5).
%C A177065 Cf. comment of _Reinhard Zumkeller_ in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 64*A002061(n+1) - 49. - _Bruno Berselli_, Aug 24 2010
%H A177065 Vincenzo Librandi, <a href="/A177065/b177065.txt">Table of n, a(n) for n = 0..1000</a>
%H A177065 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A177065 a(n) = 128*n + a(n-1) with n > 0, a(0)=15.
%F A177065 a(n) = A125169(A016754(n) - 1). - _Reinhard Zumkeller_, Jul 05 2010
%F A177065 a(0)=15, a(1)=143, a(2)=399, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Mar 13 2013
%F A177065 G.f.: (15+98*x+15*x^2)/(1-x)^3. - _Vincenzo Librandi_, Apr 08 2013
%F A177065 From _Amiram Eldar_, Feb 19 2023: (Start)
%F A177065 a(n) = A017101(n)*A004770(n).
%F A177065 Sum_{n>=0} 1/a(n) = (sqrt(2)-1)*Pi/16.
%F A177065 Sum_{n>=0} (-1)^n/a(n) = (cos(Pi/8) * log(tan(3*Pi/16)) + sin(Pi/8) * log(cot(Pi/16)))/4.
%F A177065 Product_{n>=0} (1 - 1/a(n)) = sec(Pi/8)*cos(Pi/(4*sqrt(2))).
%F A177065 Product_{n>=0} (1 + 1/a(n)) = sec(Pi/8). (End)
%F A177065 E.g.f.: exp(x)*(15 + 64*x*(2 + x)). - _Elmo R. Oliveira_, Oct 25 2024
%p A177065 A177065:=n->(8*n+3)*(8*n+5): seq(A177065(n), n=0..100); # _Wesley Ivan Hurt_, Apr 24 2017
%t A177065 Table[(8n+3)(8n+5),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{15,143,399},40] (* _Harvey P. Dale_, Mar 13 2013 *)
%t A177065 CoefficientList[Series[(15 + 98 x + 15 x^2)/(1-x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Apr 08 2013 *)
%o A177065 (Magma) [(8*n+3)*(8*n+5): n in [0..50]]; // _Vincenzo Librandi_, Apr 08 2013
%o A177065 (PARI) a(n)=(8*n+3)*(8*n+5) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A177065 Cf. A002061, A004770, A016754, A017101, A125169, A177059.
%K A177065 nonn,easy
%O A177065 0,1
%A A177065 _Vincenzo Librandi_, May 31 2010
%E A177065 Edited by _N. J. A. Sloane_, Jun 22 2010