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A337301
Triangle read by rows in which row n lists the closest integers to diagonal lengths of regular n-gon with unit edge length, n >= 4.
1
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 4, 4, 3, 3, 2, 2, 3, 3, 4, 4, 4, 3, 3, 2, 2, 3, 3, 4, 4, 4, 4, 3, 3, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 3, 4, 4, 5, 5, 5, 5, 4, 4, 3, 2, 2, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 2
OFFSET
4,2
FORMULA
T(n,k) = round(sin((k+1)*Pi/n)/sin(Pi/n)), n >= 4, 1 <= k <= n-3.
EXAMPLE
Triangle begins:
1;
2, 2;
2, 2, 2;
2, 2, 2, 2;
2, 2, 3, 2, 2;
2, 3, 3, 3, 3, 2;
2, 3, 3, 3, 3, 3, 2;
2, 3, 3, 4, 4, 3, 3, 2;
2, 3, 3, 4, 4, 4, 3, 3, 2;
2, 3, 3, 4, 4, 4, 4, 3, 3, 2;
2, 3, 4, 4, 4, 4, 4, 4, 4, 3, 2;
2, 3, 4, 4, 5, 5, 5, 5, 4, 4, 3, 2;
2, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 2;
...
Row n lists the closest integers to the length of the diagonals drawn from a fixed vertex of a regular n-gon with unit edge length, n >= 4.
The lengths of the diagonals drawn from vertex A of a regular 8-gon ABCDEFGH with unit edge length are:
AC = 1.84775...
AD = 2.41421...
AE = 2.61312...
AF = 2.41421...
AG = 1.84775...
So the row for n=8 is 2, 2, 3, 2, 2.
MATHEMATICA
T[n_, k_]:=Round[Sin[(k+1)*Pi/n]/Sin[Pi/n]]; Flatten[Table[T[n, k], {n, 4, 16}, {k, 1, n-3}]] (* Stefano Spezia, Sep 07 2020 *)
CROSSREFS
Cf. A064313.
Decimal expansion of diagonal lengths of regular n-gons with unit edge length:
n=4 A002193.
n=5 A001622.
n=9 A332437.
n=11 A231186.
Sequence in context: A309120 A098708 A067394 * A076925 A100961 A263206
KEYWORD
easy,nonn,tabl
AUTHOR
Mohammed Yaseen, Aug 22 2020
STATUS
approved