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