A296910 a(0)=1, a(1)=4; thereafter a(n) = 4*n-2*(-1)^n.

1, 4, 6, 14, 14, 22, 22, 30, 30, 38, 38, 46, 46, 54, 54, 62, 62, 70, 70, 78, 78, 86, 86, 94, 94, 102, 102, 110, 110, 118, 118, 126, 126, 134, 134, 142, 142, 150, 150, 158, 158, 166, 166, 174, 174, 182, 182, 190, 190, 198, 198, 206, 206, 214, 214, 222, 222, 230, 230, 238, 238

Offset: 0

Author

N. J. A. Sloane, Dec 22 2017

Keywords

• nonn
• easy

Comments

Coordination sequence for the bew tiling with respect to a point where two hexagons meet at only a single point. The coordination sequence for the other type of point can be shown to be A008574.

Notes: There is one point on the positive x-axis at edge-distance n from the origin iff n is even; there is one point on the positive y-axis at edge-distance n from the origin iff n>1 is odd; and the number of points inside the first quadrant at distance n from 0 is n if n is odd, and n-1 if n is even.

Then a(n) = 2*(number on positive x-axis + number on positive y-axis) + 4*(number in interior of first quadrant).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
• Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
• Reticular Chemistry Structure Resource (RCSR), The bew tiling (or net)
• Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
• N. J. A. Sloane, Illustration of initial terms. Points in the first quadrant are marked with their edge-distance from the origin (the heavy black circle). (Ignore the black rectangles, which show some fundamental cells for this tiling.)
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Apart from first two terms, same as A168384.

Cf. A008574. See A296911 for partial sums.

Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.

Programs

Mathematica

{1, 4}~Join~Array[4 # - 2 (-1)^# &, 59, 2] (* or *)
LinearRecurrence[{1, 1, -1}, {1, 4, 6, 14, 14}, 61] (* or *)
CoefficientList[Series[(1 + 3 x + x^2 + 5 x^3 - 2 x^4)/((1 - x)^2*(1 + x)), {x, 0, 60}], x] (* Michael De Vlieger, Dec 23 2017 *)

PARI

Vec((1 + 3*x + x^2 + 5*x^3 - 2*x^4) / ((1 - x)^2*(1 + x)) + O(x^50)) \\ Colin Barker, Dec 23 2017

Formula

From Colin Barker, Dec 23 2017: (Start)

G.f.: (1 + 3*x + x^2 + 5*x^3 - 2*x^4) / ((1 - x)^2*(1 + x)).

a(n) = a(n-1) + a(n-2) - a(n-3) for n>4.

(End)