A255876 a(n) = (4*n^2 + 4*n - 3 - 3*(-1)^n)/2.

4, 9, 24, 37, 60, 81, 112, 141, 180, 217, 264, 309, 364, 417, 480, 541, 612, 681, 760, 837, 924, 1009, 1104, 1197, 1300, 1401, 1512, 1621, 1740, 1857, 1984, 2109, 2244, 2377, 2520, 2661, 2812, 2961, 3120, 3277, 3444, 3609, 3784, 3957, 4140, 4321, 4512, 4701

Offset

1,1

Comments

Take an n X n square grid and add unit squares along each side except for the corners --> do this repeatedly along each side with the same restriction until no squares can be added. a(n) gives the number of vertices in each figure (see example and cf. A255840).

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

G.f.: x*(3*x^3 - 6*x^2 - x - 4)/((x + 1)*(x - 1)^3).

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

a(n) = A000290(n+1) + 4*A198442(n).

Example

                                                                 _
                                                               _|_|_
                            _              _ _               _|_|_|_|_
                          _|_|_          _|_|_|_           _|_|_|_|_|_|_
              _ _       _|_|_|_|_      _|_|_|_|_|_       _|_|_|_|_|_|_|_|_
    _        |_|_|     |_|_|_|_|_|    |_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
   |_|       |_|_|       |_|_|_|      |_|_|_|_|_|_|       |_|_|_|_|_|_|_|
                           |_|          |_|_|_|_|           |_|_|_|_|_|
                                          |_|_|               |_|_|_|
                                                                |_|
   n=1        n=2          n=3             n=4                  n=5

Maple

A255876:=n->(4*n^2 + 4*n - 3 - 3*(-1)^n)/2: seq(A255876(n), n=1..50);

Mathematica

CoefficientList[Series[(3 x^3 - 6 x^2 - x - 4)/((x + 1) (x - 1)^3), {x, 0, 50}], x]
LinearRecurrence[{2, 0, -2, 1}, {4, 9, 24, 37}, 60] (* Harvey P. Dale, Dec 26 2024 *)

Prog

(Magma) [(4*n^2 + 4*n - 3 - 3*(-1)^n)/2 : n in [1..50]];
(PARI) vector(100, n, (4*n^2 + 4*n - 3 - 3*(-1)^n)/2) \\ Derek Orr, Mar 09 2015

Crossrefs

Cf. A000290 (squares), A085046, A198442, A255840.

Sequence in context: A288103 A286729 A159068 * A158141 A056575 A056032

Adjacent sequences: A255873 A255874 A255875 * A255877 A255878 A255879

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 08 2015

Status

approved