A251629 -id:A251629 - cp's OEIS frontend

A251633 Number of lattice points of the Archimedean tiling (4,8,8) on the circles R(n) = sqrt(A251629(n) + A251631(n)*sqrt(2)) around any lattice point. First differences of A251632.

1, 3, 1, 4, 6, 2, 2, 4, 5, 4, 1, 6, 2, 4, 2, 4, 2, 2, 4, 8, 4, 4, 3, 2, 2, 1, 2, 4, 4, 2, 4, 4, 4, 8, 2, 2, 2, 8, 4, 2, 2, 4, 2, 4, 2, 1, 4, 4, 2, 1, 8, 4, 4, 2, 4, 2, 2, 4, 2, 2, 4, 4, 2, 2, 4, 2, 8, 6, 4, 6, 4, 4, 1, 8, 4, 2, 2, 1, 4, 4, 2

Offset: 0

Wolfdieter Lang, Jan 02 2015

The squares of the increasing radii of the lattice point hitting circles for the Archimedean tiling (4,8,8) are given in A251629 and A251631 as integers in Q(sqrt(2)).
For the elementary cell of the lattice we use the vectors vec(e1) from [0, 0] to [1 + sqrt(2), 0] and vec(e2) from [0, 0] to [0, 1 + sqrt(2)]. The 'atoms' in this cell are P0 = [0, 0], P1 = [0, 1], P2 = [sqrt(2)/2, 1 + sqrt(2)/2] and P3 [1  + sqrt(2)/2, 1 + sqrt(2)/2] with corresponding vectors vec(Pj), j = 0, 1, 2, 3. The general lattice point Pklj has vector vec(Pklj) = vec(Pj) + k*vec(e1) + l*vec(e2), with integer k and l.
For details see the link in A251632.

			n = 3: on the circle with R(3) = sqrt(2 + sqrt(2)), approximately 1.84776, around any lattice point lie a(3) = 4 points, namely the ones with Cartesian coordinates [+/-(sqrt(2)/2), 1  + sqrt(2)/2] and [+/-(1 + sqrt(2)/2), -sqrt(2)/2].
  The x- and y-coordinates of lattice points are obtained from the elementary cell given above.

Cf. A251629, A251631, A251632, A251628 (tiling (3,4,6,4)).

a(n) = A251632(n) - A251632(n-1), for n >= 1 and a(0) = 1.

A251631 Irrational parts of the Q(sqrt(2)) integers giving the squared radii of the lattice point circles for the Archimedean tiling (4,8,8).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 2, 3, 4, 4, 4, 6, 6, 7, 6, 8, 8, 8, 9, 10, 10, 11, 12, 12, 12, 12, 14, 13, 15, 14, 16, 16, 16, 18, 18, 18, 20, 20, 21, 20, 20, 21, 22, 22, 22, 24, 24, 25, 26, 24, 26, 27, 28, 26, 29, 30, 30, 31, 32, 30, 32, 31, 32, 32, 34, 32, 34

Offset: 0

Wolfdieter Lang, Jan 02 2015

The rational parts are found in A251629.

See the comments, examples and a link in A251629 for details. The squared radii R2(n) for lattice point hitting circles centered at any of the lattice points of the Archimedean tiling (4,8,8) are integers in the real quadratic number field Q(sqrt(2)), namely R2(n) = A251629(n) + a(n)*sqrt(2), n >= 0.

			See A251629.

Cf. A251629.

A251632 Circular disk sequence for the lattice of the Archimedean tiling (4,8,8).

Original entry on oeis.org

1, 4, 5, 9, 15, 17, 19, 23, 28, 32, 33, 39, 41, 45, 47, 51, 53, 55, 59, 67, 71, 75, 78, 80, 82, 83, 85, 89, 93, 95, 99, 103, 107, 115, 117, 119, 121, 129, 133, 135, 137, 141, 143, 147, 149, 150, 154, 158, 160, 161, 169, 173, 177, 179, 183, 185, 187, 191, 193, 195, 199, 203, 205, 207, 211, 213

Offset: 0

Wolfdieter Lang, Jan 02 2015

For the squares of the radii of the lattice point hitting circles of the Archimedean tiling (4,8,8) see A251629 and A251631.
The first differences for this sequence are given in A251633.
See the link for more details.

			n=4: The radius of the disk is R(4) = sqrt(3 + 2*sqrt(2)), approximately 2.4142. The lattice points for this R(4)-disk are the origin, three points on the circle with radius R(1) = 1, one point on the circle with radius R(2) = sqrt(2), four points on the circle with radius R(3) = sqrt(2 + sqrt(2)) and 6 points on the circle with radius R(4) = sqrt(3 + 2*sqrt(2)), all together 1 + 3 + 1 + 4 + 6 = 15 = a(4) lattice points.

Wolfdieter Lang, On lattice point circles for the Archimedean tiling (4,8,8).
Wikipedia, Archimedean tilings

Cf. A251629, A251631, A251633.

a(n) is the number of lattice points of the Archimedean tiling (4,8,8) on the boundary and the interior of the circular disk belonging to the radius R(n) = sqrt(A251629(n) + A251631(n)*sqrt(2)), for n >= 0.

cp's OEIS Frontend

A251633 Number of lattice points of the Archimedean tiling (4,8,8) on the circles R(n) = sqrt(A251629(n) + A251631(n)*sqrt(2)) around any lattice point. First differences of A251632.

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A251631 Irrational parts of the Q(sqrt(2)) integers giving the squared radii of the lattice point circles for the Archimedean tiling (4,8,8).

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A251632 Circular disk sequence for the lattice of the Archimedean tiling (4,8,8).

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