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A342235 Coordination sequence of David Eppstein's "Tetrastix" graph. 1
1, 4, 12, 28, 50, 76, 110, 148, 194, 244, 302, 364, 434, 508, 590, 676, 770, 868, 974, 1084, 1202, 1324, 1454, 1588, 1730, 1876, 2030, 2188, 2354, 2524, 2702, 2884, 3074, 3268, 3470, 3676, 3890, 4108, 4334, 4564, 4802, 5044, 5294, 5548, 5810, 6076, 6350, 6628 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The graph in question is the subgraph of the adjacency graph of Z^3, induced by the set of vertices whose coordinates are not all of the same parity. The graph is regular of degree 4, and is vertex-transitive, so it only has one vertex coordination sequence. Conjecture: the second differences are 5, 8, 6, [4, 8], where the bracketed portion repeats.

LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..250

David Eppstein, Mathstodon post about this graph, Feb 21 2021.

Rémy Sigrist, PARI program for A342235

EXAMPLE

The vertices at distance 2 from (1,0,0) are (0,-1,0),(0,0,-1),(0,0,1),(0,1,0),(1,-2,0),(1,0,-2),(1,0,2),(1,2,0),(2,-1,0),(2,0,-1),(2,0,1),(2,1,0). There are 12 of them, so a(2) = 12.

PROG

(Haskell)

-- The main entry is the last one.

-- Rotate a triple

rot :: Int -> (Int, Int, Int) -> (Int, Int, Int)

rot 0 (x, y, z) = (x, y, z)

rot 1 (y, z, x) = (x, y, z)

rot 2 (z, x, y) = (x, y, z)

-- Eppstein graph neighbors helper

egcn' :: Int -> Int -> Int -> Int -> [(Int, Int, Int)]

egcn' x y z r =

  if mod (y+z) 2 == 0

  then []

  else map (rot r) [(x-1, y, z), (x+1, y, z)]

-- Eppstein graph neighbors

egcn :: (Int, Int, Int) -> [(Int, Int, Int)]

egcn (x, y, z) =

  egcn' x y z 0 ++

  egcn' y z x 1 ++

  egcn' z x y 2

-- Eppstein graph coordination step helper

egcstep' :: [(Int, Int, Int)] -> [(Int, Int, Int)] -> [(Int, Int, Int)] -> [(Int, Int, Int)]

egcstep' _ [] next = next

egcstep' prev (this:rest) next =

  egcstep' prev rest (next ++ filter (\p -> not (elem p prev || elem p next))

                                     (egcn this))

-- Eppstein graph coordination step

egcstep :: [(Int, Int, Int)] -> [(Int, Int, Int)] -> [(Int, Int, Int)]

egcstep prev curr = egcstep' prev curr []

-- Eppstein graph circle iterator

egciter :: Int -> [(Int, Int, Int)] -> [(Int, Int, Int)] -> [(Int, Int, Int)]

egciter 0 prev curr = curr

egciter n prev curr = egciter (n - 1) curr (egcstep prev curr)

-- Eppstein graph circle; points at distance n

egc :: Int -> [(Int, Int, Int)]

egc n = egciter n [] [(1, 0, 0)]

-- Eppstein graph coordination sequence; main function.

egcs :: Int -> Int

egcs = length . egc

(PARI) See Links section.

CROSSREFS

Cf. A091999.

Sequence in context: A301005 A178571 A278211 * A192736 A109629 A112087

Adjacent sequences:  A342232 A342233 A342234 * A342236 A342237 A342238

KEYWORD

nonn

AUTHOR

Allan C. Wechsler, Mar 06 2021

EXTENSIONS

More terms from Rémy Sigrist, Mar 07 2021

STATUS

approved

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Last modified September 28 00:46 EDT 2021. Contains 347698 sequences. (Running on oeis4.)