A321019 Coordination sequence for 3-D tiling (Cairo tiling) X Z, with respect to a 5-valent point.

1, 5, 16, 36, 63, 98, 143, 196, 255, 322, 399, 484, 575, 674, 783, 900, 1023, 1154, 1295, 1444, 1599, 1762, 1935, 2116, 2303, 2498, 2703, 2916, 3135, 3362, 3599, 3844, 4095, 4354, 4623, 4900, 5183, 5474, 5775, 6084, 6399, 6722, 7055, 7396, 7743, 8098, 8463, 8836, 9215, 9602, 9999

Offset: 0

Author

N. J. A. Sloane, Nov 10 2018

Keywords

• nonn
• easy

Comments

By (Cairo tiling) X Z is meant a stack of layers of the planar Cairo tiling at integer levels.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Crossrefs

Cf. A040000, A296368.

Programs

PARI

Vec((1 + x)*(1 + x + 4*x^2 + 2*x^4 + x^5 - x^6) / ((1 - x)^3*(1 + x^2)) + O(x^50)) \\ Colin Barker, Nov 11 2018

Formula

G.f.: (x^6-x^5-2*x^4-4*x^2-x-1)*(1+x)/((x^2+1)*(x-1)^3), which is the product of the g.f.'s for A296368 and A040000.

From Colin Barker, Nov 11 2018: (Start)

a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>5.

a(n) = (-2 + (-i)^(1+n) + i^(1+n) + 8*n^2) / 2 for n>2, where i=sqrt(-1).

(End)