cp's OEIS Frontend

A379241 Maximum number of connections for a 2 X n rectangle.

%I A379241 #23 Dec 23 2024 01:57:58
%S A379241 1,5,22,54,134,269,534,934,1618,2573,4062,6030,8902,12549,17614,23854,
%T A379241 32194,42229,55238
%N A379241 Maximum number of connections for a 2 X n rectangle.
%C A379241 In a 2 X n board (with n > 1) with numbers 1 and 2, at least 2 of each, find the arrangement with more solutions connecting a pair of numbers 1 and a pair of number 2, covering the entire board and without passing through the same square twice.
%C A379241 Terms a(7)-a(20) from Giorgio Vecchi.
%C A379241 In a vertex-colored graph, partition the vertices into paths of positive lengths. It is required that the two terminal vertices of each path have the same color, and that there is exactly one such path for each color of the terminal vertices. a(n) is the maximum number of such partitions for all possible 2-colorings (at least two vertices of each color) of the 2 X n grid graph. - _Pontus von Brömssen_, Dec 19 2024
%H A379241 Rodolfo Kurchan and Claudio Meller, <a href="https://www.puzzlefun.online/problems">Number Connections</a>, Puzzle Fun, Problems (2024).
%e A379241 For n = 3 the with the board
%e A379241   +---+---+---+
%e A379241   | 1 | 1 | 2 |
%e A379241   +---+---+---+
%e A379241   | 1 | 2 | 2 |
%e A379241   +---+---+---+
%e A379241 the maximum number of solutions is 5 being the squares with this letters:
%e A379241   +---+---+---+
%e A379241   | A | B | C |
%e A379241   +---+---+---+
%e A379241   | D | E | F |
%e A379241   +---+---+---+
%e A379241 Solutions:
%e A379241   1) ABED - CF
%e A379241   2) ADEB - CF
%e A379241   3) BAD - CFE
%e A379241   4) AD - CBEF
%e A379241   5) AD - EBCF
%e A379241 There are five solutions so a(3) = 5.
%K A379241 nonn,more
%O A379241 2,2
%A A379241 _Rodolfo Kurchan_, Dec 18 2024