A296910 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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).

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)

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 *)

PROG

(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