A372855 Number of ways two dihexes can be placed on an n-th regular hexagonal board.

0, 33, 702, 3630, 11409, 27603, 56748, 104352, 176895, 281829, 427578, 623538, 880077, 1208535, 1621224, 2131428, 2753403, 3502377, 4394550, 5447094, 6678153, 8106843, 9753252, 11638440, 13784439, 16214253, 18951858, 22022202, 25451205, 29265759, 33493728

Offset

1,2

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = (3/2)*(27*n^4 - 90*n^3 + 78*n^2 + 11*n - 24), for n > 1.

a(n) = 5*a(n - 1) - 10*a(n - 2) + 10*a(n - 3) - 5*a(n - 4) + a(n - 5) for n > 6.

G.f.: 3*x^2*(11 + 179*x + 150*x^2 - 17*x^3 + x^4)/(1 - x)^5.

E.g.f.: 36 - 3*x + 3*exp(x)*(27*x^4 + 72*x^3 - 3*x^2 + 26*x - 24)/2. - Stefano Spezia, Jun 04 2024

Example

Regular hexagonal boards n = 1...4:
. ___
./   \
.\___/
.     ___
. ___/   \___
./   \___/   \
.\___/   \___/
./   \___/   \
.\___/   \___/
.    \___/
.         ___
.     ___/   \___
. ___/   \___/   \___
./   \___/   \___/   \
.\___/   \___/   \___/
./   \___/   \___/   \
.\___/   \___/   \___/
./   \___/   \___/   \
.\___/   \___/   \___/
.    \___/   \___/
.        \___/
.             ___
.         ___/   \___
.     ___/   \___/   \___
. ___/   \___/   \___/   \___
./   \___/   \___/   \___/   \
.\___/   \___/   \___/   \___/
./   \___/   \___/   \___/   \
.\___/   \___/   \___/   \___/
./   \___/   \___/   \___/   \
.\___/   \___/   \___/   \___/
./   \___/   \___/   \___/   \
.\___/   \___/   \___/   \___/
.    \___/   \___/   \___/
.        \___/   \___/
.            \___/
For n = 2 the a(2) = 33: (without grid)
. . . . . . . . . . . . . . . . . . .
.   x---x   .   x---x   .   x---x   .
.           .           .           .
. x---x   o . o   x---x . o   o   o .
.           .           .           .
.   o   o   .   o   o   .   x---x   .
. . . . . . . . . . . . . . . . . . .
.   x---x   .   x---x   .   x---x   .
.           .           .           .
. x   o   o . o   x   o . o   x   o .
.  \        .      \    .    /      .
.   x   o   .   o   x   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   x---x   .   o   o   .   o   x   .
.           .           .        \  .
. o   o   x . x---x   o . x---x   x .
.        /  .           .           .
.   o   x   .   x---x   .   o   o   .
. . . . . . . . . . . . . . . . . . .
.   o   o   .   o   o   .   o   o   .
.           .           .           .
. x---x   x . o   x---x . x   x---x .
.        /  .           .  \        .
.   o   x   .   x---x   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   x   o   .   o   x   .
.  /        .    \      .        \  .
. x   x---x . o   x   o . o   o   x .
.           .           .           .
.   o   o   .   x---x   .   x---x   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   o   x   .   x   x   .
.  /        .      /    .    \   \  .
. x   o   o . o   x   o . o   x   x .
.           .           .           .
.   x---x   .   x---x   .   o   o   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   x   o   .   o   x   .
.    \      .    \      .        \  .
. x   x   o . o   x   x . x   o   x .
.  \        .        /  .  \        .
.   x   o   .   o   x   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   o   x   .   x   x   .   o   x   .
.        \  .  /     \  .        \  .
. o   x   x . x   o   x . o   x   x .
.      \    .           .    /      .
.   o   x   .   o   o   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   o   o   .   o   x   .   o   o   .
.           .      /    .           .
. x   x   o . x   x   o . x   o   x .
.  \   \    .  \        .  \     /  .
.   x   x   .   x   o   .   x   x   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   x   x   .   x   o   .
.  /        .  /   /    .  /        .
. x   x   o . x   x   o . x   x   o .
.      \    .           .    /      .
.   o   x   .   o   o   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   o   x   .   o   o   .
.  /        .      /    .           .
. x   o   x . o   x   x . o   x   x .
.        /  .        /  .    /   /  .
.   o   x   .   o   x   .   x   x   .
. . . . . . . . . . . . . . . . . . .

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 33, 702, 3630, 11409, 27603}, 50] (* Paolo Xausa, Aug 28 2024 *)

Crossrefs

Cf. A000384, A242856.

Sequence in context: A028178 A028153 A028105 * A028103 A028144 A028098

Adjacent sequences: A372852 A372853 A372854 * A372856 A372857 A372858

Keyword

nonn,easy

Author

Nicolas Bělohoubek, May 15 2024

Status

approved