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

0,3

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.

Links

Table of n, a(n) for n=0..48.

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

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

Example

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

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]

Prog

(Magma) [(4*n^2 - 4*n + 1 - (-1)^n)/2 : n in [0..100]];
(PARI) vector(100, n, (4*(n-1)^2 - 4*(n-1) + 1 + (-1)^n)/2) \\ Derek Orr, Mar 09 2015

Crossrefs

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

Cf. A056640, A085046, A255876.

Sequence in context: A112263 A244291 A085018 * A001741 A272702 A272734

Adjacent sequences: A255837 A255838 A255839 * A255841 A255842 A255843

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 07 2015

Status

approved