A259486 a(n) = 3*n^2 - 3*n + 1 + 6*floor((n-1)*(n-2)/6).

1, 7, 19, 43, 73, 109, 157, 211, 271, 343, 421, 505, 601, 703, 811, 931, 1057, 1189, 1333, 1483, 1639, 1807, 1981, 2161, 2353, 2551, 2755, 2971, 3193, 3421, 3661, 3907, 4159, 4423, 4693, 4969, 5257, 5551, 5851, 6163, 6481, 6805, 7141, 7483, 7831, 8191, 8557

Offset: 1

Author

Wesley Ivan Hurt, Jun 28 2015

Keywords

• nonn
• easy

Comments

Start with the geometric picture for the centered hex numbers (A003215). Here, each hexagonal figure in the sequence is the aggregate of smaller unit hexes (with n hexes along each side). Then, when possible, add additional unit hexes to each side except for the corners --> do this repeatedly with the same restriction until no hexes can be added. a(n) gives the area of each figure (see example).

a(n) == 1 mod 6. - Robert Israel, Jun 29 2015

Examples

			-----------------------------------------------------------------------
Figure 1
-----------------------------------------------------------------------
                                                  __    __    __
                                                 /  \__/  \__/  \
                                                 \_*/  \__/  \*_/
                              __               __/  \__/  \__/  \__
                           __/  \__           /  \__/  \__/  \__/  \
            __          __/  \__/  \__        \__/  \__/  \__/  \__/
         __/  \__      /  \__/  \__/  \     __/  \__/  \__/  \__/  \__
.__     /  \__/  \     \__/  \__/  \__/    / *\__/  \__/  \__/  \__/* \
/  \    \__/  \__/     /  \__/  \__/  \    \__/  \__/  \__/  \__/  \__/
\__/    /  \__/  \     \__/  \__/  \__/       \__/  \__/  \__/  \__/
        \__/  \__/     /  \__/  \__/  \       /  \__/  \__/  \__/  \
           \__/        \__/  \__/  \__/       \__/  \__/  \__/  \__/
                          \__/  \__/             \__/  \__/  \__/
                             \__/                / *\__/  \__/* \
                                                 \__/  \__/  \__/
n=1         n=2               n=3                       n=4
-----------------------------------------------------------------------
Table 1
-----------------------------------------------------------------------
a(1) = 1                              =  1
a(2) = 3  + 2(2)                      =  7
a(3) = 5  + 2(3+4)                    =  19
a(4) = 7  + 2(4+5+6)          + 6(1)  =  43
a(5) = 9  + 2(5+6+7+8)        + 6(2)  =  73
a(6) = 11 + 2(6+7+8+9+10)     + 6(3)  =  109
a(7) = 13 + 2(7+8+9+10+11+12) + 6(5)  =  157
...
		

Links

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

Crossrefs

Cf. A003215 (hex numbers), A000969, A130518, A255840 (similar, with squares).

Programs

Magma

[3*n^2 - 3*n + 1 + 6*Floor((n-1)*(n-2)/6) : n in [1..100]];

Magma

I:=[1,7,19,43,73]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-3)-2*Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jul 14 2015

Maple

A259486:=n->3*n^2 - 3*n + 1 + 6*floor((n-1)*(n-2)/6): seq(A259486(n), n=1..100);

Mathematica

Table[3 n^2 - 3 n + 1 + 6 Floor[(n - 1) (n - 2)/6], {n, 50}] (* or *)
CoefficientList[Series[(1 + 5 x + 6 x^2 + 11 x^3 + x^4)/((1 - x)^3 (1 + x + x^2)), {x, 0, 50}], x]
LinearRecurrence[{2, -1, 1, -2, 1}, {1, 7, 19, 43, 73}, 50]; (* Vincenzo Librandi, Jul 14 2015 *)

Formula

G.f.: (1+5*x+6*x^2+11*x^3+x^4)/((1-x)^3*(1+x+x^2)).

a(n) = 2*a(n-1)-a(n-2)+a(n-3)-2*a(n-4)+a(n-5), n>5.

a(n) = A003215(n+1) + 6*A130518(n+1).

From Robert Israel, Jun 29 2015: (Start)

a(n) = 4*n^2 - 6*n + 1 if 3 divides n, 4*n^2 - 6*n + 3 otherwise.

a(n) = 1 + 6 * A000969(n-2) for n >= 2. (End)

a(n) = 4*n^2 - 6*n + 3^sign(n mod 3). - Wesley Ivan Hurt, Jul 13 2015