A386479 a(0) = 0; thereafter a(n) = 2*n^2 - 3*n + 5.
0, 4, 7, 14, 25, 40, 59, 82, 109, 140, 175, 214, 257, 304, 355, 410, 469, 532, 599, 670, 745, 824, 907, 994, 1085, 1180, 1279, 1382, 1489, 1600, 1715, 1834, 1957, 2084, 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
Offset: 0
Links
- N. J. A. Sloane, 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.
- N. J. A. Sloane, Illustration for a(3) = 14. Two 3-chains can divide the plane into at most 14 regions.
- N. J. A. Sloane, Two 4-chains can divide the plane into at most a(4) = 25 regions. (The two 4-chains are colored respectively black and green.)
- N. J. A. Sloane, Two 5-chains can divide the plane into at most a(5) = 40 regions. (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.
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Crossrefs
Programs
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Mathematica
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 *)
Formula
From Stefano Spezia, Jul 26 2025: (Start)
G.f.: -x*(4-5*x+5*x^2) / (x-1)^3.
E.g.f.: exp(x)*(5 - x + 2*x^2) - 5. (End)
Extensions
Changed a(0) so as to match changes to A386478. - N. J. A. Sloane, Jul 26 2025
Comments