This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A283240 #18 Feb 16 2025 08:33:42 %S A283240 2,2,0,4,8,8,0,0,4,12,20,16,0,0,0,4,16,32,40,28,0,0,0,0,4,20,48,72,72, %T A283240 44,0,0,0,0,0,4,24,68,120,144,120,64,0,0,0,0,0,0,4,28,92,188,264,264, %U A283240 188,88,0,0,0,0,0,0,0,4,32,120,280,452,528,452,280,116 %N A283240 Triangle read by rows: T(n,k) = number of directed self-avoiding walks (SAWs) of length k in an n-ladder graph that use all rows of the graph. %C A283240 n is the number of rows in the ladder graph, i.e., L_n. %C A283240 k is the length of the directed SAWs. k = 0 represents the single nodes with no edges. %C A283240 T(n,k) is the number of directed SAWs which use at least one node from every row. %H A283240 OEIS, <a href="https://oeis.org/wiki/Self-avoiding_walks">Self-avoiding walks</a>. %H A283240 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ladder_graph">Ladder graph</a>. %H A283240 Wolfram MathWorld, <a href="https://mathworld.wolfram.com/LadderGraph.html">Ladder Graph</a>. %F A283240 T(n,k) = 0 when k+1 < n %F A283240 T(n,k) = 4 when k+1 = n %F A283240 T(n,k) = 2(n^2-n+2) when k = 2n-1 %F A283240 T(n,k) = T(n-1,k-1) + T(n-1,k-2) + 4 when k = 2n-2 %F A283240 T(n,k) = T(n-1,k-1) + T(n-1,k-2) otherwise %e A283240 Triangle T(n,k) begins: %e A283240 n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 %e A283240 1 2 2 %e A283240 2 0 4 8 8 %e A283240 3 0 0 4 12 20 16 %e A283240 4 0 0 0 4 16 32 40 28 %e A283240 5 0 0 0 0 4 20 48 72 72 44 %e A283240 6 0 0 0 0 0 4 24 68 120 144 120 64 %e A283240 7 0 0 0 0 0 0 4 28 92 188 264 264 188 88 %e A283240 8 0 0 0 0 0 0 0 4 32 120 280 452 528 452 280 116 %e A283240 9 0 0 0 0 0 0 0 0 4 36 152 400 732 980 980 732 400 148 %e A283240 10 0 0 0 0 0 0 0 0 0 4 40 188 552 1132 1712 1960 1712 1132 552 184 %e A283240 e.g., there are T(3,3) = 12 directed SAWs of length 3 in L_3 that use at least one node from each row. %e A283240 Six shapes walked in both directions. %e A283240 > _ _ %e A283240 > | | | | |_ _| %e A283240 > | |_ | _| | | %o A283240 (Python) %o A283240 maxN=20 %o A283240 Tnk=[[0 for column in range(0, row*2)] for row in range(0,maxN+1)] %o A283240 Tnk[1][0]=2 # initial values for the special case of 1-ladder. Two single nodes. %o A283240 Tnk[1][1]=2 # SAW of length 1 on a L_1, either left or right %o A283240 for row in range(2,maxN+1): %o A283240 for column in range(0, row*2): %o A283240 if(column+1 < row): %o A283240 # path is smaller than ladder - no possible SAW using all rows %o A283240 Tnk[row][column] = 0 %o A283240 elif(column+1 == row): %o A283240 # vertical SAW, 2 possible in 2 directions %o A283240 Tnk[row][column] = 4 %o A283240 elif(column == row*2 -1): %o A283240 # n-ladder Hamiltonian A137882 %o A283240 Tnk[row][column] = 2*(row*row - row + 2) %o A283240 elif(column == 2*(row-1)): %o A283240 # Grow SAW including Hamiltonians from previous row, 4 extra SAWs from Hamiltonians %o A283240 Tnk[row][column] = Tnk[row-1][column-1] + Tnk[row-1][column-2] + 4 %o A283240 else: %o A283240 # Grow SAW from previous SAWs. Either adding one or two edges %o A283240 Tnk[row][column] = Tnk[row-1][column-1] + Tnk[row-1][column-2] %o A283240 print(Tnk) %Y A283240 The sum of each columns is twice A038577. %Y A283240 The diagonal T(n,2n-1) are the number of directed Hamiltonian paths in the n-ladder graph A137882. %Y A283240 Primarily used to calculate the total number of directed SAWs of length k in an n-ladder A283241. %K A283240 nonn,tabf,walk %O A283240 1,1 %A A283240 _Hector J. Partridge_, Mar 03 2017