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A259486 a(n) = 3*n^2 - 3*n + 1 + 6*floor((n-1)*(n-2)/6). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

Table of n, a(n) for n=1..47.

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

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

EXAMPLE

-----------------------------------------------------------------------

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

...

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

PROG

(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

CROSSREFS

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

Sequence in context: A145993 A265676 A054690 * A298034 A054691 A139828

Adjacent sequences:  A259483 A259484 A259485 * A259487 A259488 A259489

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Jun 28 2015

STATUS

approved

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Last modified August 1 16:54 EDT 2021. Contains 346400 sequences. (Running on oeis4.)