A393441 a(n) = (n^4-n^3+2*n^2-2*n+2)/2.

1, 1, 7, 34, 109, 271, 571, 1072, 1849, 2989, 4591, 6766, 9637, 13339, 18019, 23836, 30961, 39577, 49879, 62074, 76381, 93031, 112267, 134344, 159529, 188101, 220351, 256582, 297109, 342259, 392371, 447796, 508897, 576049, 649639, 730066, 817741, 913087, 1016539, 1128544, 1249561, 1380061, 1520527, 1671454, 1833349, 2006731, 2192131, 2390092

Offset

0,3

Comments

a(n) is the maximum number of regions that can be formed in the plane by drawing n n-armed V's. This is the main diagonal of A386481.

Links

Table of n, a(n) for n=0..47.

David O. H. Cutler, Jonas Karlsson, and Neil J. A. Sloane, Cutting a Pancake with an Exotic Knife, arXiv:2511.15864[math.CO], v3, April 19 2026.

N. J. A. Sloane, Illustration for a(3) = 34.

N. J. A. Sloane, Illustration for a(4) = 109.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

From Enrique Navarrete, Mar 26 2026: (Start)

G.f.: (4*x^4 - x^3 + 12*x^2 - 4*x + 1)/(1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

E.g.f.: (1/2)*(x^4 + 5*x^3 + 6*x^2 + 2)*exp(x).

a(n) = A092365(n) - (n-1), n > 0. (End)

Mathematica

a[n_]:=(n^4-n^3+2*n^2-2*n+2)/2; Array[a, 48, 0] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 1, 7, 34, 109, 271}, 48] (* or *)
CoefficientList[Series[(-1+4x-12x^2+x^3-4x^4)/(-1+x)^5, {x, 0, 47}], x] (* James C. McMahon, Mar 25 2026 *)

Crossrefs

Cf. A386481, A386478, A387525.

Cf. A092365.

Sequence in context: A259055 A195018 A024817 * A201230 A071598 A224036

Adjacent sequences: A393438 A393439 A393440 * A393442 A393443 A393444

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 25 2026

Status

approved