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 A016814 #66 Jan 25 2024 07:33:00
%S A016814 1,25,81,169,289,441,625,841,1089,1369,1681,2025,2401,2809,3249,3721,
%T A016814 4225,4761,5329,5929,6561,7225,7921,8649,9409,10201,11025,11881,12769,
%U A016814 13689,14641,15625,16641,17689,18769,19881,21025,22201,23409,24649,25921,27225,28561,29929
%N A016814 a(n) = (4*n + 1)^2.
%C A016814 A bisection of A016754. Sequence arises from reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the squares A000290. - _Omar E. Pol_, May 24 2008
%H A016814 G. C. Greubel, <a href="/A016814/b016814.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..200 from Ivan Panchenko).
%H A016814 Leo Tavares, <a href="/A016814/a016814.jpg">Illustration: Square trapeziums</a>
%H A016814 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A016814 a(n) = a(n-1) + 32*n - 8, n > 0. - _Vincenzo Librandi_, Dec 15 2010
%F A016814 From _George F. Johnson_, Sep 28 2012: (Start)
%F A016814 G.f.: (1 + 22*x + 9*x^2)/(1 - x)^3.
%F A016814 a(n+1) = a(n) + 16 + 8*sqrt(a(n)).
%F A016814 a(n+1) = 2*a(n) - a(n-1) + 32 = 3*a(n) - 3*a(n-1) + a(n-2).
%F A016814 a(n-1)*a(n+1) = (a(n) - 16)^2 ; a(n+1) - a(n-1) = 16*sqrt(a(n)).
%F A016814 a(n) = A016754(2*n) = (A016813(n))^2. (End)
%F A016814 Sum_{n>=0} 1/a(n) = G/2 + Pi^2/16, where G is the Catalan constant (A006752). - _Amiram Eldar_, Jun 28 2020
%F A016814 Product_{n>=1} (1 - 1/a(n)) = 2*Gamma(5/4)^2/sqrt(Pi) = 2 * A068467^2 * A087197. - _Amiram Eldar_, Feb 01 2021
%F A016814 From _G. C. Greubel_, Dec 28 2022: (Start)
%F A016814 a(2*n) = A017078(n).
%F A016814 a(2*n+1) = A017126(n).
%F A016814 E.g.f.: (1 + 24*x + 16*x^2)*exp(x). (End)
%F A016814 a(n) = A272399(n+1) - A014105(n). - _Leo Tavares_, Dec 24 2023
%p A016814 A016814:=n->(4*n+1)^2; seq(A016814(k), k=0..100); # _Wesley Ivan Hurt_, Nov 02 2013
%t A016814 (4*Range[0,40] +1)^2 (* or *) LinearRecurrence[{3,-3,1}, {1,25,81}, 40] (* _Harvey P. Dale_, Nov 20 2012 *)
%t A016814 Accumulate[32Range[0, 47] - 8] + 9 (* _Alonso del Arte_, Aug 19 2017 *)
%o A016814 (PARI) a(n)=(4*n+1)^2 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A016814 (Magma) [(4*n+1)^2: n in [0..40]]; // _G. C. Greubel_, Dec 28 2022
%o A016814 (SageMath) [(4*n+1)^2 for n in range(41)] # _G. C. Greubel_, Dec 28 2022
%Y A016814 Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m-3), this sequence (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).
%Y A016814 Cf. A001539, A006752, A016742, A016802, A016813.
%Y A016814 Cf. A016826, A016838, A017126, A068467, A087197.
%Y A016814 Cf. A272399, A014105.
%K A016814 nonn,easy
%O A016814 0,2
%A A016814 _N. J. A. Sloane_