A316317 Coordination sequence for trivalent node in chamfered version of square grid.

1, 3, 6, 11, 14, 15, 20, 25, 26, 29, 34, 37, 40, 43, 46, 51, 54, 55, 60, 65, 66, 69, 74, 77, 80, 83, 86, 91, 94, 95, 100, 105, 106, 109, 114, 117, 120, 123, 126, 131, 134, 135, 140, 145, 146, 149, 154, 157, 160, 163, 166, 171, 174, 175, 180, 185, 186, 189, 194

Offset

0,2

Links

Rémy Sigrist, Table of n, a(n) for n = 0..5000

Michel Deza and Mikhail Shtogrin, Isometric embedding of mosaics into cubic lattices, Discrete mathematics 244.1-3 (2002): 43-53. See Fig. 2.

Michel Deza and Mikhail Shtogrin, Isometric embedding of mosaics into cubic lattices, Discrete mathematics 244.1-3 (2002): 43-53. [Annotated scan of page 52 only]

Rémy Sigrist, PARI program for A316317

Rémy Sigrist, Illustration of first terms

N. J. A. Sloane, Initial terms of coordination sequence for trivalent node

N. J. A. Sloane, "Basketweave" tiling by 3X1 rectangles which is equivalent (as far as the graph and coordination sequences are concerned) to this tiling

Formula

Apparently, a(n + 12) = a(n) + 40 for any n > 0. - Rémy Sigrist, Jun 30 2018

This can surely be proved by the Coloring Book Method, although I have not worked out the details. See A316316 for the corresponding proof for a tetravalent node. - N. J. A. Sloane, Jun 30 2018

G.f. (assuming above conjecture): (1+x)^2*(1+3*x^2+x^4)/((1-x)^2*(1+x+x^2)*(1+x^2)). - Robert Israel, Jul 01 2018

a(n) = (30*n - 9*A056594(n-1) + 6*A102283(n))/9 for n > 0. - Conjectured by Stefano Spezia, Jun 12 2021

Prog

(PARI) See Links section.

Crossrefs

See A316316 for tetravalent node.

See A250120 for links to thousands of other coordination sequences.

Cf. A316358 (partial sums).

Cf. A056594, A102283.

Sequence in context: A316096 A310091 A136981 * A074737 A310092 A190439

Adjacent sequences: A316314 A316315 A316316 * A316318 A316319 A316320

Keyword

nonn

Author

N. J. A. Sloane, Jun 29 2018

Extensions

More terms from Rémy Sigrist, Jun 30 2018

Status

approved