A381109 a(n) = (21*n^2 + 9*n + 2)/2.

1, 16, 52, 109, 187, 286, 406, 547, 709, 892, 1096, 1321, 1567, 1834, 2122, 2431, 2761, 3112, 3484, 3877, 4291, 4726, 5182, 5659, 6157, 6676, 7216, 7777, 8359, 8962, 9586, 10231, 10897, 11584, 12292, 13021, 13771, 14542, 15334, 16147, 16981, 17836, 18712

Offset

0,2

Comments

A 9-sided figure is obtained by truncating every other corner of a centered hexagonal figure as in A003215; a(n) is the number of dots in such a 9-sided figure with n+1 dots on each side.

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 71.

Links

Table of n, a(n) for n=0..42.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

From Stefano Spezia, Oct 19 2025: (Start)

G.f.: (1 + 13*x + 7*x^2)/(1 - x)^3.

E.g.f.: exp(x)*(2 + 30*x + 21*x^2)/2. (End)

Example

Illustration of initial terms:
                                            o o o o
                                           o o o o o o
                       o o o              o o o o o o o o
                      o o o o o          o o o o o o o o o o
         o o         o o o o o o o        o o o o o o o o o o
        o o o o       o o o o o o o      o o o o o o o o o o o
  o      o o o o     o o o o o o o o      o o o o o o o o o o o
        o o o o       o o o o o o o      o o o o o o o o o o o
         o o         o o o o o o o        o o o o o o o o o o
                      o o o o o          o o o o o o o o o o
                       o o o              o o o o o o o o
                                           o o o o o o
                                            o o o o
  1         16             52                    109

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 16, 52}, 43] (* James C. McMahon, Mar 07 2025 *)

Crossrefs

Cf. A003215, A005892, A381424.

Sequence in context: A044499 A241489 A009953 * A009939 A009935 A241490

Adjacent sequences: A381106 A381107 A381108 * A381110 A381111 A381112

Keyword

nonn,easy

Author

Aaron David Fairbanks, Mar 06 2025

Status

approved