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 A185669 #40 Sep 08 2022 08:45:55
%S A185669 2,9,24,47,78,117,164,219,282,353,432,519,614,717,828,947,1074,1209,
%T A185669 1352,1503,1662,1829,2004,2187,2378,2577,2784,2999,3222,3453,3692,
%U A185669 3939,4194,4457,4728,5007,5294,5589,5892,6203,6522,6849,7184,7527,7878,8237,8604,8979,9362,9753,10152,10559,10974,11397,11828
%N A185669 a(n) = 4*n^2 + 3*n + 2.
%C A185669 Natural numbers A000027 written clockwise as a square spiral:
%C A185669 .
%C A185669 43--44--45--46--47--48--49
%C A185669 |
%C A185669 42 21--22--23--24--25--26
%C A185669 | | |
%C A185669 41 20 7---8---9--10 27
%C A185669 | | | | |
%C A185669 40 19 6 1---2 11 28
%C A185669 | | | | | |
%C A185669 39 18 5---4---3 12 29
%C A185669 | | | |
%C A185669 38 17--16--15--14--13 30
%C A185669 | |
%C A185669 37--36--35--34--33--32--31
%C A185669 .
%C A185669 Walking in straight lines away from the center:
%C A185669 1, 2, 11, ... = A054552(n) = 1 -3*n+4*n^2,
%C A185669 1, 8, 23, ... = A033951(n) = 1 +3*n+4*n^2,
%C A185669 1, 3, 13, ... = A054554(n+1) = 1 -2*n-4*n^2,
%C A185669 1, 7, 21, ... = A054559(n+1) = 1 +2*n+4*n^2,
%C A185669 1, 4, 15, ... = A054556(n+1) = 1 -n+4*n^2,
%C A185669 1, 6, 19, ... = A054567(n+1) = 1 +n+4*n^2,
%C A185669 1, 5, 17, ... = A053755(n) = 1 +4*n^2,
%C A185669 1, 9, 25, ... = A016754(n) = 1 +4*n+4*n^2 = (1+2*n)^2,
%C A185669 2, 8, 22, ... = 2*A084849(n) = 2 +2*n+4*n^2,
%C A185669 2, 12, 30, ... = A002939(n+1) = 2 +6*n+4*n^2,
%C A185669 2, 9, 24, ... = a(n) = 2 +3*n+4*n^2,
%C A185669 2, 10, 26, ... = A069894(n) = 2 +4*n+4*n^2,
%C A185669 3, 11, 27, ... = A164897(n) = 3 +4*n+4*n^2,
%C A185669 3, 12, 29, ... = A054552(n+1)+1 = 3 +5*n+4*n^2,
%C A185669 3, 14, 33, ... = A033991(n+1) = 3 +7*n+4*n^2,
%C A185669 3, 15, 35, ... = A000466(n+1) = 3 +8*n+4*n^2,
%C A185669 4, 14, 32, ... = 2*A130883(n+1) = 4 +6*n+4*n^2,
%C A185669 4, 16, 36, ... = A016742(n+1) = 4 +8*n+4*n^2 = (2+2*n)^2,
%C A185669 5, 18, 39, ... = A007742(n+1) = 5 +9*n+4*n^2,
%C A185669 5, 19, 41, ... = A125202(n+2) = 5+10*n+4*n^2.
%H A185669 Ivan Panchenko, <a href="/A185669/b185669.txt">Table of n, a(n) for n = 0..1000</a>
%H A185669 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A185669 a(n) = a(n-1) + 8*n - 1.
%F A185669 a(n) = 2*a(n-1) - a(n-2) + 8.
%F A185669 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A185669 G.f.: (2 +3*x +3*x^2)/(1-x)^3 . - _R. J. Mathar_, Feb 11 2011
%F A185669 a(n) = A033954(n) + 2. - _Bruno Berselli_, Apr 10 2011
%F A185669 E.g.f.: (4*x^2 + 7*x + 2)*exp(x). - _G. C. Greubel_, Jul 09 2017
%t A185669 Table[4n^2 + 3n + 2, {n,0,50}] (* _G. C. Greubel_, Jul 09 2017 *)
%t A185669 LinearRecurrence[{3,-3,1},{2,9,24},60] (* _Harvey P. Dale_, Aug 11 2021 *)
%o A185669 (Magma) [2+3*n+4*n^2: n in [0..80]]; // _Vincenzo Librandi_, Feb 09 2011
%o A185669 (PARI) a(n)=4*n^2+3*n+2 \\ _Charles R Greathouse IV_, Apr 14 2014
%K A185669 nonn,easy
%O A185669 0,1
%A A185669 _Paul Curtz_, Feb 09 2011