A271741 -id:A271741 - cp's OEIS frontend

A268606 Number of ways to trisect a hexagon with side length n exactly into three identical parts in a triangular lattice.

Original entry on oeis.org

1, 5, 116, 15785, 11599297, 47212453928, 1100377983366327, 148568527921382084692

Offset: 1

Luca Petrone, Feb 08 2016

G. P. Jelliss, Dissected Hexagons, The Games and Puzzles Journal, Issue 22, January-April 2002.
Luca Petrone, Illustration showing a(3) = 116

Cf. A257952, A271741, A271857.

Added a(6) from Jelliss's website by Vaclav Kotesovec, Mar 03 2016

a(7)-a(8) from Andrew Howroyd, Apr 11 2016

A271857 Number of ways to dissect a hexagon with side length n exactly into six identical parts in a triangular lattice.

Original entry on oeis.org

1, 2, 12, 173, 5429, 392544, 66961869, 27094069322, 26124568587557, 60352331499840380, 335377713005955826349, 4494480789037552980419332, 145516206571394421594063628243, 11398373584242623552596178870957640, 2162546126021822830176241418936795142991

Offset: 1

Andrew Howroyd, Apr 15 2016

nonn

This sequence only enumerates those dissections in which the six pieces are arranged around the center at 60-degree intervals. Other dissections are possible, such as those formed from those trisections which can be further bisected into identical parts.

G. P. Jelliss, Dissected Hexagons, The Games and Puzzles Journal, Issue 22, January-April 2002.

Cf. A113900, A268606, A271741.

A271723 Numbers k such that 3*k - 8 is a square.

Original entry on oeis.org

3, 4, 8, 11, 19, 24, 36, 43, 59, 68, 88, 99, 123, 136, 164, 179, 211, 228, 264, 283, 323, 344, 388, 411, 459, 484, 536, 563, 619, 648, 708, 739, 803, 836, 904, 939, 1011, 1048, 1124, 1163, 1243, 1284, 1368, 1411, 1499, 1544, 1636, 1683, 1779, 1828, 1928, 1979, 2083, 2136, 2244, 2299

Offset: 1

Juri-Stepan Gerasimov, Apr 13 2016

Square roots of resulting squares gives A001651. - Ray Chandler, Apr 14 2016

			a(1) = 3 because 3*3 - 8 = 1^2.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001651.

Cf. numbers n such that 3*n + k is a square: this sequence (k=-8), A120328 (k=-6), A271713 (k=-5), A056107 (k=-3), A257083 (k=-2), A033428 (k=0), A001082 (k=1), A080663 (k=3), A271675 (k=4), A100536 (k=6), A271741 (k=7), A067725 (k=9).

Magma
```
[n: n in [1..2400] | IsSquare(3*n-8)];
```

seq(seq(((3*m+k)^2+8)/3, k=1..2),m=0..50); # Robert Israel, Dec 05 2016

Select[Range@ 2400, IntegerQ@ Sqrt[3 # - 8] &] (* Bruno Berselli, Apr 14 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{3,4,8,11,19},60] (* Harvey P. Dale, Oct 02 2020 *)

from gmpy2 import is_square
[n for n in range(3000) if is_square(3*n-8)] # Bruno Berselli, Dec 05 2016

[(6*(n-1)*n-(2*n-1)*(-1)**n+23)/8 for n in range(1, 60)] # Bruno Berselli, Dec 05 2016

From Ilya Gutkovskiy, Apr 13 2016: (Start)
G.f.: x*(3 + x - 2*x^2 + x^3 + 3*x^4)/((1 - x)^3*(1 + x)^2).
a(n) = (6*(n - 1)*n - (2*n - 1)*(-1)^n + 23)/8. (End)

A003155 Number of ways to halve an n X n chessboard.

Original entry on oeis.org

1, 1, 1, 6, 15, 255, 1897, 92263, 1972653, 281035054, 17635484470, 7490694495750, 1405083604458437, 1789509008288411290

Offset: 1

N. J. A. Sloane

This is the number of ways to cut an n X n chessboard along grid lines into two identical pieces. In the case that n is odd the central square is first removed to ensure an even number of squares. Solutions that differ only by rotation or reflection are considered equivalent. - Andrew Howroyd, Apr 19 2016

M. Gardner, The Unexpected Hanging and Other Mathematical Diversions. Simon and Schuster, NY, 1969, p. 189.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Illustration showing ways to quarter a 6x6 board

Cf. A113900, A006067, A257952, A271741.

a(2n) = A113900(n). - Andrew Howroyd, Apr 19 2016

a(10) corrected and a(11)-a(14) from Andrew Howroyd, Apr 19 2016

cp's OEIS Frontend

A268606 Number of ways to trisect a hexagon with side length n exactly into three identical parts in a triangular lattice.

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A271857 Number of ways to dissect a hexagon with side length n exactly into six identical parts in a triangular lattice.

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A271723 Numbers k such that 3*k - 8 is a square.

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A003155 Number of ways to halve an n X n chessboard.

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