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 A166147 #63 Feb 16 2025 08:33:11
%S A166147 1,17,41,73,113,161,217,281,353,433,521,617,721,833,953,1081,1217,
%T A166147 1361,1513,1673,1841,2017,2201,2393,2593,2801,3017,3241,3473,3713,
%U A166147 3961,4217,4481,4753,5033,5321,5617,5921,6233,6553,6881,7217,7561,7913,8273,8641
%N A166147 a(n) = 4*n^2 + 4*n - 7.
%C A166147 Primes in the sequence are in A028886. - _Bruno Berselli_, Mar 16 2012
%C A166147 The number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood. - _Robert Price_, May 19 2016
%C A166147 a(n) = y - x for any primitive Pythagorean triangle (x^2 + y^2 = z^2), where z - x = 8. Also, a(n+2) = y + x, and y = 8n + 12. - _Boyd Blundell_, Jul 31 2021
%D A166147 Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A166147 Vincenzo Librandi, <a href="/A166147/b166147.txt">Table of n, a(n) for n = 1..1000</a>
%H A166147 N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.
%H A166147 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>.
%H A166147 Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>.
%H A166147 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%H A166147 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>.
%H A166147 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>.
%H A166147 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>.
%F A166147 a(n) = a(n-1)+8*n with n>1, a(1)=1.
%F A166147 From _Vincenzo Librandi_, Mar 15 2012: (Start)
%F A166147 G.f.: x*(1+14*x-7*x^2)/(1-x)^3.
%F A166147 a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). (End)
%F A166147 E.g.f.: (-7 + 8*x + 4*x^2)*exp(x) + 7. - _G. C. Greubel_, Apr 26 2016
%F A166147 Sum_{n>=1} 1/a(n) = 1/7 + (Pi/(8*sqrt(2)))*tan(sqrt(2)*Pi). - _Amiram Eldar_, Feb 20 2023
%t A166147 CoefficientList[Series[(1+14x-7x^2)/(1-x)^3,{x,0,50}],x] (* or *) LinearRecurrence[{3, -3, 1}, {1, 17, 41}, 50] (* _Vincenzo Librandi_, Mar 15 2012 *)
%t A166147 Table[4 n^2 + 4 n - 7, {n, 46}] (* _Michael De Vlieger_, Apr 27 2016 *)
%o A166147 (PARI) a(n)=8*binomial(n+1,2)-7 \\ _Charles R Greathouse IV_, Jan 11 2012
%o A166147 (Magma) I:=[1, 17, 41]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Mar 15 2012
%Y A166147 Cf. A028886.
%K A166147 nonn,easy
%O A166147 1,2
%A A166147 _Vincenzo Librandi_, Oct 08 2009
%E A166147 New name from _Charles R Greathouse IV_, Jan 11 2012