A255840 a(n) = (4*n^2 - 4*n + 1 - (-1)^n)/2.

0, 1, 4, 13, 24, 41, 60, 85, 112, 145, 180, 221, 264, 313, 364, 421, 480, 545, 612, 685, 760, 841, 924, 1013, 1104, 1201, 1300, 1405, 1512, 1625, 1740, 1861, 1984, 2113, 2244, 2381, 2520, 2665, 2812, 2965, 3120, 3281, 3444, 3613, 3784, 3961, 4140, 4325, 4512

Offset: 0

Author

Wesley Ivan Hurt, Mar 07 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) is the total area of each figure. The perimeter, P, of each figure is given by P(n) = 4*A042963(n), n>0 (see example).

For n>0, partial sums of a(n) are in A056640.

Examples

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

Links

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

Crossrefs

Cf. A000290 (squares), A002620 (quarter-squares), A042963.

Cf. A056640, A085046, A255876.

Programs

Magma

[(4*n^2 - 4*n + 1 - (-1)^n)/2 : n in [0..100]];

Maple

A255840:=n->(4*n^2 - 4*n + 1 - (-1)^n)/2: seq(A255840(n), n=0..100);

Mathematica

CoefficientList[Series[x (1 + 2 x + 5 x^2)/((1 + x) (1 - x)^3), {x, 0, 50}], x]

PARI

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

Formula

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

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

a(n) = A000290(n) + 4*A002620(n).

a(n) - a(n-1) = A047471(n). - Wesley Ivan Hurt, Apr 28 2017