A383465 a(n) = 25*n^2/2 - 11*n/2 + 1.

1, 8, 40, 97, 179, 286, 418, 575, 757, 964, 1196, 1453, 1735, 2042, 2374, 2731, 3113, 3520, 3952, 4409, 4891, 5398, 5930, 6487, 7069, 7676, 8308, 8965, 9647, 10354, 11086, 11843, 12625, 13432, 14264, 15121, 16003, 16910, 17842, 18799, 19781, 20788, 21820, 22877, 23959, 25066, 26198, 27355, 28537, 29744, 30976, 32233, 33515, 34822

Offset: 0

Author

N. J. A. Sloane, Jul 16 2025

Keywords

• nonn
• easy

Comments

David O. H. Cutler (paper in preparation) defines a 5-chain to be a planar graph consisting of a continuous path made up of five straight segments. Then a(n) is the maximum number of pieces that the plane can be divided into by drawing n 5-chains.

Related to maximum number of regions that can be obtained in the plane by drawing n pentagrams.

Links

• N. J. A. Sloane, Illustration for n=1: a 5-chain that divides the plane into a(1) = 8 regions. Two of the line segments are infinite.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

A row of the array in A386478.

Programs

Mathematica

a[n_]:= 25*n^2/2 - 11*n/2 + 1;Array[a,54,0] (* or *) LinearRecurrence[{3,-3,1},{1,8,40},54] (* or *) CoefficientList[Series[(19*x^2+5*x+1)/(1-x)^3,{x,0,53}],x] (* James C. McMahon, Jul 16 2025 *)

Formula

G.f.: (19*x^2+5*x+1)/(1-x)^3. - Alois P. Heinz, Jul 16 2025