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

Author

Wolfdieter Lang, Jan 02 2015

Keywords

• nonn
• easy

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.