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

0,2

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).

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)

Prog

(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

Crossrefs

Cf. A040000, A296368.

Sequence in context: A363613 A363605 A108966 * A184635 A328506 A072333

Adjacent sequences: A321016 A321017 A321018 * A321020 A321021 A321022

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 10 2018

Status

approved