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

Author

Wesley Ivan Hurt, Mar 08 2015

Keywords

• nonn
• easy

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).

Examples

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

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).

Crossrefs

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

Programs

Magma

[(4*n^2 + 4*n - 3 - 3*(-1)^n)/2 : n in [1..50]];

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 *)

PARI

vector(100,n,(4*n^2 + 4*n - 3 - 3*(-1)^n)/2) \\ Derek Orr, Mar 09 2015

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).