A342235 Coordination sequence of David Eppstein's "Tetrastix" graph.

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

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