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A316316
Coordination sequence for tetravalent node in chamfered version of square grid.
4
1, 4, 8, 8, 12, 20, 20, 20, 28, 32, 32, 36, 40, 44, 48, 48, 52, 60, 60, 60, 68, 72, 72, 76, 80, 84, 88, 88, 92, 100, 100, 100, 108, 112, 112, 116, 120, 124, 128, 128, 132, 140, 140, 140, 148, 152, 152, 156, 160, 164, 168, 168, 172, 180, 180, 180, 188, 192, 192
OFFSET
0,2
LINKS
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]
Michel Deza and Mikhail Shtogrin, Enlargement of figure from previous link
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 on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.
N. J. A. Sloane, Trunks and branches structure of tetravalent node (First part of proof that a(n+12)=a(n)+40).
N. J. A. Sloane, Calculation of coordination sequence (Second part of proof that a(n+12)=a(n)+40).
N. J. A. Sloane, An equivalent tiling seen on the sidewalk of East 70th St in New York City. As far as the graph and coordination sequences are concerned, this is the same as the chamfered square grid. The trivalent vertices labeled b and c are equivalent to each other.
FORMULA
Apparently, a(n + 12) = a(n) + 40 for any n > 0. - Rémy Sigrist, Jun 30 2018
From N. J. A. Sloane, Jun 30 2018: This conjecture is true.
Theorem: a(n + 12) = a(n) + 40 for any n > 0.
The proof uses the Coloring Book Method described in the Goodman-Strauss - Sloane article. For details see the two links.
From Colin Barker, Dec 13 2018: (Start)
G.f.: (1 + 3*x + 5*x^2 + 2*x^3 + 5*x^4 + 3*x^5 + x^6) / ((1 - x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-1) - a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) - a(n-6) for n>6.
(End)
a(n) = (2/9)*(15*n + 9*A056594(n-1) - 6*A102283(n)) for n > 0. - Stefano Spezia, Jun 12 2021
MATHEMATICA
Join[{1}, LinearRecurrence[{1, -1, 2, -1, 1, -1}, {4, 8, 8, 12, 20, 20}, 100]] (* Jean-François Alcover, Dec 13 2018 *)
PROG
(PARI) See Links section.
CROSSREFS
See A316317 for trivalent node.
See A250120 for links to thousands of other coordination sequences.
Cf. A316357 (partial sums).
Sequence in context: A375027 A294963 A014198 * A333288 A159786 A083744
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jun 29 2018
EXTENSIONS
More terms from Rémy Sigrist, Jun 30 2018
STATUS
approved