The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A341291 a(n) = number of nodes of degree 3 or 4 that are at distance n from the origin in the graph of the Rick Kenyon tiling of the first quadrant by L and backwards-L tiles. 5
 1, 2, 4, 6, 7, 9, 11, 14, 15, 17, 19, 20, 21, 23, 24, 28, 28, 31, 32, 35, 36, 39, 40, 44, 44, 46, 48, 49, 50, 52, 54, 57, 58, 60, 62, 64, 65, 66, 68, 71, 72, 74, 76, 77, 78, 80, 81, 84, 84, 87, 88, 90, 92, 94, 96, 99, 100, 102, 104, 105, 106, 108, 110, 113 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS There is a unique way to tile the first quadrant with L-tiles and backward-L tiles. Upside down-L's are not allowed. Consider the graph formed by the 1-skeleton of the tiling, with nodes at points of degree 3 or 4. The sequence gives the number of nodes at distance 0 from the node at the origin. Proof that this tiling is unique, from Jaap Scherphuis, Feb 19 2021 (Start): Build the tiling row by row, from bottom to top, left to right. When you fill a row, you must keep the row above tileable. Since all the tiles have a length-2 base, any unfilled part of a row with no spaces underneath it is only tileable if it has even length. When you place a tile, its stalk protrudes into the row above, and only one choice of the tile makes the unfilled part to the left of the stalk have even length. (End) It appears that the number of vertical infinite "cuts" in the wall that start at height n is given by the Jacobsthal sequence A001045. - Maurizio Paolini, Feb 09 2021 REFERENCES Peter Winkler, Mathematical Puzzles, CRC Press, 2021; see p. lxx. LINKS Rémy Sigrist, Table of n, a(n) for n = 0..8191 Chaim Goodman-Strauss, Two-colored illustration of bottom six rows of tiling Rémy Sigrist, Illustration of initial terms Rémy Sigrist, Colored representation of the nodes at distance <= 511 from the origin (the color is a function of the distance) Rémy Sigrist, C++ program for A341291 FORMULA Label a grid point with a 1 if it is the left end-point of the base of a normal L tile, with a 2 if it is the right end-point of the base of a backwards L-tile (which we could call an R tile), and with a 0 otherwise. Then it appears that above the line x = 3y/2, (x,y) has label 1 if x=3m, y=2n; 2 if x=3m, y=2n+1; and 0 otherwise. It would be nice to have a formula for the points below the line. - N. J. A. Sloane, Feb 11 2021 MATHEMATICA (* Mathematica code from Chaim Goodman-Strauss for generating illustration, Feb 09 2021 *) (* setting things up *) Clear[L, R, l, r]; Protect[L, R, l, r]; (*the base of an L tile is LR; the top is either an l (over an L) or an r (over an R)*) nextRow[lastRow_]:=(row ={}; While[Length[row]

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 23 10:33 EDT 2021. Contains 345397 sequences. (Running on oeis4.)