A353447 - OEIS

A353447

a(n) is the number of tetrapods standing on the four edges of an n X n grid, so that no two feet are the same distance apart and no foot is on a corner. Tetrapods with congruent footprints are counted only once.

%I #25 Jun 11 2022 05:51:15

%S 0,0,1,11,40,105,190,379,616,987,1426,2139,2964,4130,5403,7180,9155,

%T 11716,14458,18092,22037,26808,31793,38343,45060,53184,61613,71878,

%U 82466,95368,108195,123790,140040,158457,177405,200020,223039,248769,275214,306411,337645

%N a(n) is the number of tetrapods standing on the four edges of an n X n grid, so that no two feet are the same distance apart and no foot is on a corner. Tetrapods with congruent footprints are counted only once.

%C If we name the tetrapod's footprints "mini-frame", we can say that mini-frames span their grid, i.e., there is no smaller grid for them. Every corner-less set of points with distinct distances in a smallest possible n X n grid contains at least one mini-frame.

%H Hugo Pfoertner, <a href="/A353447/b353447.txt">Table of n, a(n) for n = 3..200</a>

%e .

%e . C . a(3) = 0 . . . C .

%e D . B <=== since AB = CD . . . . .

%e . A . is forbidden . . . . B

%e . . . . .

%e . C . . D . . . .

%e a(4) = 0 ===> ? . . . . A . . .

%e (there is no ? . . B ______________

%e space for D) . A . . a(5) = 1

%e (No other solutions)

%e .

%e . . . . . The tetrapod has 6 distinct

%e D . . . . squared distances 4, 5, 10,

%e . . . . C <===== 13, 17, 18, but it uses only

%e . . . . . three edges of the 5 X 5 grid.

%e . A . B . (Not allowed.)

%e .

%Y Cf. A193838, A271490, A335232, A351699, A351700, A353532.

%Y The general case without excluding the corners of the grid rectangle is covered in A354700 and A354701.

%K nonn

%O 3,4

%A _Rainer Rosenthal_, Apr 20 2022

%E a(23) and beyond from _Hugo Pfoertner_, Apr 20 2022