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 A283241 #16 Feb 16 2025 08:33:42 %S A283241 2,2,4,8,8,8,6,14,20,28,20,16,8,20,32,52,56,64,40,28,10,26,44,76,96, %T A283241 132,128,128,72,44,12,32,56,100,136,204,240,296,264,232,120,64,14,38, %U A283241 68,124,176,276,356,492,548,608,504,392,188,88 %N A283241 Triangle read by rows: S(n,k) = total number of directed self-avoiding walks (SAWs) of length k in an n-ladder graph. %C A283241 n is the number of rows in the ladder graph, i.e., L_n. %C A283241 k is the length of the directed SAWs. k = 0 represents the single nodes with no edges. %C A283241 S(n,k) is the number of distinct SAWs of length k in the n-ladder graph. %H A283241 OEIS, <a href="https://oeis.org/wiki/Self-avoiding_walks">Self-avoiding walks</a>. %H A283241 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ladder_graph">Ladder graph</a>. %H A283241 Wolfram MathWorld, <a href="https://mathworld.wolfram.com/LadderGraph.html">Ladder Graph</a>. %F A283241 Using T(n,k) from A283240, %F A283241 S(n,k) = T(n,k) + 2*T(n,k-1) + 3*T(n,k-2) + 4*T(n,k-4) + ... + (k+1)*T(n,0) %e A283241 Triangle S(n,k) begins: %e A283241 n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 %e A283241 1 2 2 %e A283241 2 4 8 8 8 %e A283241 3 6 14 20 28 20 16 %e A283241 4 8 20 32 52 56 64 40 28 %e A283241 5 10 26 44 76 96 132 128 128 72 44 %e A283241 6 12 32 56 100 136 204 240 296 264 232 120 64 %e A283241 7 14 38 68 124 176 276 356 492 548 608 504 392 188 88 %e A283241 8 16 44 80 148 216 348 472 692 864 1104 1168 1172 904 628 280 116 %e A283241 9 18 50 92 172 256 420 588 892 1184 1636 1984 2352 2352 2148 1540 964 400 148 %e A283241 10 20 56 104 196 296 492 704 1092 1504 2172 2840 3720 4352 4800 4512 3772 2512 1428 552 184 %e A283241 e.g., there are T(3,3) = 28 directed SAWs of length 3 in L_3. %e A283241 Fourteen shapes walked in both directions: %e A283241 > _ _ %e A283241 > | | | | |_ _| %e A283241 > | |_ | _| | | %e A283241 > _ . . _ _ . . . . %e A283241 > |_| . . | | _ |_ _| _ _ %e A283241 > . . |_| . . | | . . . . |_ _| %o A283241 (Python) %o A283241 # As A283240 but Tnk initialized as grid instead of a triangle %o A283241 maxN=20 %o A283241 Tnk=[[0 for column in range(0, maxN*2)] for row in range(0, maxN+1)] %o A283241 Tnk[1][0]=2 # initial values for the special case of 1-ladder. Two single nodes. %o A283241 Tnk[1][1]=2 # SAW of length 1 on a L_1, either left or right %o A283241 for row in range(2, maxN+1): %o A283241 for column in range(0, row*2): %o A283241 if(column+1 < row): %o A283241 # path is smaller than ladder - no possible SAW using all rows %o A283241 Tnk[row][column] = 0 %o A283241 elif(column+1 == row): %o A283241 # vertical SAW, 2 possible in 2 directions %o A283241 Tnk[row][column] = 4 %o A283241 elif(column == row*2 -1): %o A283241 # n-ladder Hamiltonian A137882 %o A283241 Tnk[row][column] = 2*(row*row - row + 2) %o A283241 elif(column == 2*(row-1)): %o A283241 # Grow SAW including Hamiltonians from previous row, 4 extra SAWs from Hamiltonians %o A283241 Tnk[row][column] = Tnk[row-1][column-1] + Tnk[row-1][column-2] + 4 %o A283241 else: %o A283241 # Grow SAW from previous SAWs. Either adding one or two edges %o A283241 Tnk[row][column] = Tnk[row-1][column-1] + Tnk[row-1][column-2] %o A283241 # Sum multiples of the columns above this one e.g. T(n,k) + 2T(n,k-1) + 3T(n,k-2) + ... %o A283241 Snk=[[0 for column in range(0,row*2)] for row in range(0,maxN+1)] %o A283241 for row in range(1,maxN+1): %o A283241 for column in range(0,(row*2)): %o A283241 for i in range(0,row): %o A283241 Snk[row][column]+=(i+1)*Tnk[row-i][column] %o A283241 print(Snk) %Y A283241 Uses A283240 for the number of n-ladders with no empty rows. %Y A283241 The right diagonal is the number of directed Hamiltonian paths in the n-ladder graph (A137882). %K A283241 nonn,tabf,walk %O A283241 1,1 %A A283241 _Hector J. Partridge_, Mar 03 2017