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 A386479 #69 Jul 30 2025 04:40:30
%S A386479 0,4,7,14,25,40,59,82,109,140,175,214,257,304,355,410,469,532,599,670,
%T A386479 745,824,907,994,1085,1180,1279,1382,1489,1600,1715,1834,1957,2084,
%U A386479 2215,2350,2489,2632,2779,2930,3085,3244,3407,3574,3745,3920,4099,4282,4469,4660,4855,5054,5257,5464,5675,5890,6109,6332,6559,6790
%N A386479 a(0) = 0; thereafter a(n) = 2*n^2 - 3*n + 5.
%C A386479 For n>0, a(n) is the maximum number of regions the plane can be divided into by drawing two n-chains (both finite and infinite regions are counted). See A386478 for further information.
%C A386479 We do not at present have an explicit construction that will achieve a(n) for n > 5.
%H A386479 N. J. A. Sloane, <a href="/A386479/a386479_2.jpg">Two 1-chains (i.e., lines) can divide the plane into at most a(1) = 4 regions; two 2-chains (i.e., V's) can divide the plane into at most a(2) = 7 regions.</a>
%H A386479 N. J. A. Sloane, <a href="/A386479/a386479_1.jpg">Illustration for a(3) = 14.</a> Two 3-chains can divide the plane into at most 14 regions.
%H A386479 N. J. A. Sloane, <a href="/A386479/a386479_3.jpg">Two 4-chains can divide the plane into at most a(4) = 25 regions.</a> (The two 4-chains are colored respectively black and green.)
%H A386479 N. J. A. Sloane, <a href="/A386479/a386479_4.jpg">Two 5-chains can divide the plane into at most a(5) = 40 regions.</a> (The two 5-chains are colored respectively black and red.) It would be nice to have a clearer picture. Region 36 is tiny. Also some of the points where arms of the 5-chains meet are just outside the region shown.
%H A386479 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A386479 From _Stefano Spezia_, Jul 26 2025: (Start)
%F A386479 G.f.: -x*(4-5*x+5*x^2) / (x-1)^3.
%F A386479 E.g.f.: exp(x)*(5 - x + 2*x^2) - 5. (End)
%t A386479 LinearRecurrence[{3,-3,1},{5,4,7},60] (* or *) a[n_]:=2n^2-3n+5;Array[a,60,0] (* _James C. McMahon_, Jul 26 2025 *)
%Y A386479 A column of the array in A386478.
%Y A386479 Essentially the same (up to offset, initial terms, and the addition of a small constant) as several other sequences, including A014105, A014107, A084849, A096376, A236257, ....
%K A386479 nonn,easy
%O A386479 0,2
%A A386479 _N. J. A. Sloane_, Jul 25 2025
%E A386479 Changed a(0) so as to match changes to A386478. - _N. J. A. Sloane_, Jul 26 2025