A373036
Triangle read by rows: T(n,k) is the number of edge subsets E of the n X k grid graph such that E contains a path between the top left node and the bottom right node, 1 <= k <= n.
Original entry on oeis.org
1, 1, 7, 1, 40, 1135, 1, 216, 28942, 3329245, 1, 1144, 707239, 358911148, 167176484530, 1, 6016, 16963938, 37502829018, 74568672196498, 140386491543732211, 1, 31552, 403490839, 3856945416544, 32485805235240376, 256258754970108999490, 1946586793700869420041631
Offset: 1
Triangle begins:
1;
1, 7;
1, 40, 1135;
1, 216, 28942, 3329245;
1, 1144, 707239, 358911148, 167176484530;
1, 6016, 16963938, 37502829018, 74568672196498, 140386491543732211;
...
A349594
Number of 2 X n mazes that can be navigated from the top left corner to the bottom right corner.
Original entry on oeis.org
1, 7, 40, 216, 1144, 6016, 31552, 165312, 865792, 4533760, 23739904, 124305408, 650874880, 3408031744, 17844699136, 93436084224, 489237741568, 2561682178048, 13413142233088, 70232124948480, 367740181282816, 1925512588951552, 10082114810675200, 52790638512439296
Offset: 1
For n = 2 the a(2) = 7 solutions are as follows:
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
| | | | | | | | | | | | | | | |
+ + + + + + + +---+ + + + +---+ + +---+ + + +---+
| | | | | | | | | | | | | | | |
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
-
import Data.List
m = [[2, 0, 2], [0, 2, 2], [1, 1, 4]]
(.*.) :: Num a => [[a]] -> [[a]] -> [[a]]
(.*.) a b = [[ sum $ zipWith (*) ar bc | bc <- (transpose b)] | ar <- a ]
(.^.) :: Num a => [[a]] -> Integer -> [[a]]
m .^. 0 = [ [ if i == j then 1 else 0 | i <- [1 .. n] ] | j <- [1 .. n] ] where n = length m
m .^. n | even n = let m' = m .^. (n `div` 2) in m' .*. m'
| otherwise = m .*. (m .^. (n - 1))
a349594 n = (z !! 0 !! 1) + (z !! 0 !! 2) + (z !! 2 !! 1) + (z !! 2 !! 2) where z = m .^. (n - 1)
-
a:= n-> (<<0|1|0>, <0|0|1>, <8|-16|8>>^n. <<0, 1, 7>>)[1, 1]:
seq(a(n), n=1..24); # Alois P. Heinz, Dec 09 2021
-
LinearRecurrence[{8, -16, 8}, {1, 7, 40}, 24] (* Jean-François Alcover, Jan 29 2025 *)
-
Vec((1 - x)/((1 - 2*x)*(1 - 6*x + 4*x^2)) + O(x^30)) \\ Andrew Howroyd, Nov 22 2021
A365629
Number of 4 X n mazes that can be navigated from the top left corner to the bottom right corner.
Original entry on oeis.org
1, 216, 28942, 3329245, 358911148, 37502829018, 3856945416544, 393396697543644, 39951066751274152, 4047887027105625168, 409638762069161924728, 41428094401248851559736, 4188336537335577744595384, 423360539638841208001947048, 42789587016771330584001089176
Offset: 1
- Pontus von Brömssen, Table of n, a(n) for n = 1..499
- Eugene Nonko, Description of the problem and the approach to calculation.
- Eugene Nonko, C program.
- Steven B. Segletes, On the Electrical Connectivity of a 2-D, Randomly Distributed, Two-Component (Conducting/Insulating) Mixture, Devcom, ARL-TR-8899, Jan 2020; page 15 lists a(4).
- Index entries for linear recurrences with constant coefficients, order 26.
Showing 1-3 of 3 results.
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