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A245618
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Triangle {H(n,k)} similar to Pascal's with sides of 1's, but interior entries are obtained by the rule: H(n,k) = |H(n-1,k)+(-1)^m(n,k)*H(n-1,k-1)|, where m(n,k) = H(n-1,k) + H(n-1,k-1).
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5
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1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 4, 4, 1, 1, 1, 2, 3, 8, 3, 2, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 2, 2, 6, 10, 6, 2, 2, 1, 1, 1, 4, 8, 16, 16, 8, 4, 1, 1, 1, 2, 3, 12, 24, 32, 24, 12, 3, 2, 1, 1, 1, 1, 9, 36, 56, 56, 36, 9, 1, 1, 1, 1, 2, 2, 10
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OFFSET
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0,5
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COMMENTS
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Let us consider the operation <+> over integers such that k<+>m = |k+(-1)^(k+m)*m|. Then H(n,k) = H(n-1,k)<+>H(n-1,k-1).
This is an analog of the formula binomial(n,k) = binomial(n-1,k) + binomial(n-1,k-1).
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LINKS
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EXAMPLE
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Triangle begins
1
1 1
1 2 1
1 1 1 1
1 2 2 2 1
1 1 4 4 1 1
1 2 3 8 3 2 1
....................
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MATHEMATICA
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parityAdd[a_, b_] := Abs[a + b (-1)^(a + b)];
triangleHP[n_, 0] := 1;
triangleHP[n_, n_] := 1;
triangleHP[n_, k_] := triangleHP[n, k] = parityAdd[triangleHP[n - 1, k - 1], triangleHP[n - 1, k]];
Flatten[Table[triangleHP[n, k], {n, 0, 15}, {k, 0, n}]] (* Peter J. C. Moses, Nov 05 2014 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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