cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A249870 Rational parts of the Q(sqrt(3)) integers giving the square of the radii for lattice point circles for the Archimedean tiling (3, 4, 6, 4).

Original entry on oeis.org

0, 1, 2, 3, 2, 4, 4, 4, 5, 6, 8, 8, 7, 8, 10, 10, 10, 13, 14, 11, 12, 13, 15, 14, 16, 16, 17, 16, 19, 20, 22, 19, 20, 20, 24, 23, 21, 25, 22, 23, 28, 26, 26, 28, 31, 28, 32, 28, 28, 30, 32, 34, 35, 32, 33, 38, 34, 36, 38, 37, 40, 37, 38, 43, 40, 44, 40, 46
Offset: 0

Views

Author

Wolfdieter Lang, Dec 06 2014

Keywords

Comments

The irrational parts are given in A249871.
The points of the lattice of the Archimedean tiling (3, 4, 6, 4) lie on certain circles around any point. The length of the side of the regular 6-gon is taken as 1 (in some length unit).
The squares of the radii R2(n) of these circles are integers in the real quadratic number field Q(sqrt(3)), hence R2(n) = a(n) + A249871(n)*sqrt(3). The R2 sequence is sorted in increasing order.
For details see the notes given in a link.
This computation was inspired by a construction given by Kival Ngaokrajang in A245094.

Examples

			The pairs [a(n), A249871(n)] for the squares of the radii R2(n) begin:
[0, 0], [1, 0], [2, 0], [3, 0], [2, 1], [4, 0], [4, 1], [4, 2], [5, 2], [6, 3], [8, 2], [8, 3], [7, 4], [8, 4], [10, 3], ...
The corresponding radii R(n) are (Maple 10 digits, if not an integer):
0, 1, 1.414213562, 1.732050808, 1.931851653, 2, 2.394170171, 2.732050808, 2.909312911, 3.346065215, 3.385867927, 3.632650881, 3.732050808, 3.863703305, 3.898224265 ...

Links

Crossrefs

Cf. A249871, A251627, A251628.

A249871 Irrational parts of the Q(sqrt(3)) integers giving the square of the radii for lattice point circles for the Archimedean tiling (3,4,6,4).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 2, 2, 3, 2, 3, 4, 4, 3, 4, 5, 4, 4, 6, 6, 6, 6, 8, 7, 8, 8, 9, 8, 8, 8, 10, 10, 11, 9, 10, 12, 10, 12, 12, 10, 12, 13, 13, 12, 14, 12, 15, 16, 15, 15, 15, 16, 18, 18, 16, 19, 18, 17, 18, 18, 20, 20, 18, 20, 18, 21, 19, 22
Offset: 0

Views

Author

Wolfdieter Lang, Dec 06 2014

Keywords

Comments

The corresponding rational parts are given in A249870.
The square of the radii R2(n) of lattice point circles around any lattice point of the Archimedean tiling (3,4,6,4) are integers in Q(sqrt(3)): R2(n) = A249870(n) + a(n)*sqrt(3), n >= 0. They are sorted in increasing order. For details, especially the coordinates of the lattice points, see the note given in a link under A249870.

Examples

			See A249870.

Crossrefs

Cf. A249870, A251627, A251628.

A251627 Circular disk sequence for the lattice of the Archimedean tiling (3,4,6,4).

Original entry on oeis.org

1, 5, 7, 9, 13, 14, 18, 25, 29, 33, 35, 39, 43, 45, 49, 51, 55, 57, 59, 63, 69, 73, 77, 79, 83, 89, 93, 97, 99, 101, 103, 107, 109, 113, 117, 121, 123, 127, 129, 133, 134, 136, 140, 144, 146, 158, 160, 164, 165, 169, 173, 177, 181, 183, 187
Offset: 0

Views

Author

Wolfdieter Lang, Dec 09 2014

Keywords

Comments

For the squares of the radii of the lattice point hitting circles of the Archimedean tiling (3,4,6,4) see A249870 and A249871.
The first differences for this sequence are given in A251628.

Examples

			n=4: The radius of the disk is R(4) = sqrt(2 + sqrt(3)), approximately 1.932. The lattice points for this R(n)-disk are the origin, four points on the circle with radius R(1) = 1, two points on the circle with radius R(2) = sqrt(2), two points on the circle with radius R(3) = sqrt(3) and 4 points on the circle with radius R(4) = sqrt(2+sqrt(3)), all together 1 + 4 + 2 + 2 + 4 = 13 = a(4) lattice points. See Figure 3 of the note given in the link.

Links

Crossrefs

Cf. A249870, A249871, A251628.

Formula

a(n) is the number of lattice points of the Archimedean tiling (3,4,6,4) on the boundary and the interior of the circular disk belonging to the radius R(n) = sqrt(A249870(n) + A249871(n)* sqrt(3)), for n >= 0.

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.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Jan 02 2015

Keywords

Comments

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.

Examples

			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.

Crossrefs

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

Formula

a(n) = A251632(n) - A251632(n-1), for n >= 1 and a(0) = 1.
Showing 1-4 of 4 results.

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