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 A174579 Number of spanning trees in the n-triangular grid graph. 1
 1, 3, 54, 5292, 2723220, 7242690816, 98719805835000, 6861326937782575104, 2423821818614367091537296, 4342290918217084382837760000000, 39389085041906366256386454778172877408, 1807026244113880332171608161401397806958116864 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The n-triangular grid graph has n+1 rows with k vertices in row k. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The Graph has A000217(n+1) vertices and 3*A000217(n) edges altogether. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..50 Eric Weisstein's World of Mathematics, Spanning Tree Eric Weisstein's World of Mathematics, Triangular Grid Graph Wikipedia, Kirchhoff's theorem MAPLE with(LinearAlgebra): tr:= n-> n*(n+1)/2: a:= proc(n) local h, i, M;       if n=0 then 1 else         M:= Matrix(tr(n+1), shape=symmetric);         for h in [seq(seq([[i, i+j], [i, i+j+1], [i+j, i+j+1]][],                            i=tr(j-1)+1 .. tr(j-1)+j), j=1..n)]         do M[h[]]:= -1 od;         for i to tr(n+1) do M[i, i]:= -add(M[i, j], j=1..tr(n+1)) od;         Determinant(DeleteColumn(DeleteRow(M, 1), 1))       fi     end: seq(a(n), n=0..12); MATHEMATICA tr[n_] := n*(n + 1)/2; a[0] = 1; a[n_] := Module[{T, M}, T = Table[Table[{{i, i+j}, {i, i+j+1}, {i + j, i+j+1}}, {i, tr[j-1]+1, tr[j-1] + j}], {j, 1, n}] // Flatten[#, 2]&; M = Array[0&, {tr[n+1], tr[n+1]}]; Do[{i, j} = h; M[[i, j]] = -1, {h, T}]; M = M + Transpose[M]; For[i = 1, i <= tr[n+1], i++, M[[i, i]] = -Sum[M[[i, j]], {j, 1, tr[n+1]}]]; Det[Rest /@ Rest[M]]]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Jun 02 2018, from Maple *) CROSSREFS Cf. A000217. Sequence in context: A003027 A054545 A158103 * A171739 A157568 A156911 Adjacent sequences:  A174576 A174577 A174578 * A174580 A174581 A174582 KEYWORD nonn AUTHOR Alois P. Heinz, Nov 29 2010 EXTENSIONS Indexing changed by Alois P. Heinz, Jun 14 2017 STATUS approved

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Last modified May 27 05:24 EDT 2020. Contains 334649 sequences. (Running on oeis4.)