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A108038
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Triangle read by rows: g.f. = (x+y+x*y)/((1-x-x^2)*(1-y-y^2)).
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3
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0, 1, 1, 1, 3, 1, 2, 4, 4, 2, 3, 7, 5, 7, 3, 5, 11, 9, 9, 11, 5, 8, 18, 14, 16, 14, 18, 8, 13, 29, 23, 25, 25, 23, 29, 13, 21, 47, 37, 41, 39, 41, 37, 47, 21, 34, 76, 60, 66, 64, 64, 66, 60, 76, 34, 55, 123, 97, 107, 103, 105, 103, 107, 97, 123, 55, 89, 199, 157, 173, 167, 169, 169, 167
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OFFSET
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0,5
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COMMENTS
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Start with 3 rows 0; 1 1; 1 3 1; then rule is each entry is maximum of sum of two entries diagonally above it to the left or to the right. Borders are Fibonacci numbers (A000045).
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LINKS
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FORMULA
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T(n,k) = F(k+2)*F(n-k+2) - F(k+1)*F(n-k+1), where F(n) = Fibonacci(n) = A000045(n).
T(n,k) = F(k)*F(n-k+2) + F(k+1)*F(n-k), where F(n) = Fibonacci(n).
T(n,k) = T(n-1,k) + T(n-2,k) and T(n,k) = T(n-1,k-1) + T(n-2,k-2), where T(1,1) = 0, T(2,1) = T(2,2) = 1, and T(3,2) = 3.
G.f: (x + x*y + x^2*y)/((1 - x - x^2)*(1 - x*y - x^2*y^2)). (End)
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EXAMPLE
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Triangle begins:
k=0 1 2 3 4
n=0: 0;
n=1: 1, 1;
n=2: 1, 3, 1;
n=3: 2, 4, 4, 2;
n=4: 3, 7, 5, 7, 3;
...
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MATHEMATICA
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Block[{nn = 11, s}, s = Series[(x + y + x*y)/((1 - x - x^2)*(1 - y - y^2)), {x, 0, nn}, {y, 0, nn}]; Table[Function[m, SeriesCoefficient[s, {m, k}]][n - k], {n, 0, nn}, {k, 0, n}]] // Flatten (* Michael De Vlieger, Dec 04 2020 *)
G[n_, k_] := Fibonacci[k]*Fibonacci[n-k+1]; T[n_, k_]:= G[n+2, k+1]-G[n, k]; RowPointHosoya[n_] := Table[Inset[T[n, i+1], {1-n+2i, 1-n}], {i, 0, n-1}]; T[n_] := Graphics[ Flatten[Table[RowPointHosoya[i], {i, 1, n}], 1]]; Manipulate[T[n], Style["Determinant Hosoya Triangle", 12, Red], {{n, 6, "Rows"}, Range[12]}, ControlPlacement -> Up] (* Rigoberto Florez, Feb 07 2022 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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