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A143656
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Triangle T(n, k) = A045545(k) if gcd(n,k) = 1, 0 otherwise, read by rows.
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2
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1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 2, 3, 0, 1, 0, 0, 0, 7, 0, 1, 1, 2, 3, 7, 8, 0, 1, 0, 2, 0, 7, 0, 22, 0, 1, 1, 0, 3, 7, 0, 22, 32, 0, 1, 0, 2, 0, 0, 0, 22, 0, 66, 0, 1, 1, 2, 3, 7, 8, 22, 32, 66, 91, 0, 1, 0, 0, 0, 7, 0, 22, 0, 0, 0, 233, 0, 1, 1, 2, 3, 7, 8, 22, 32, 66, 91, 233, 263, 0
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OFFSET
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1,9
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COMMENTS
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Sum of row terms = A045545 starting with offset 1: (1, 1, 2, 3, 7, 8, 22,...).
A045545 also = rightmost diagonal with nonzero terms.
Sum of n-th row terms = rightmost nonzero term of next row.
Prime n rows = first (n-1) terms of (1, 1, 2, 3, 7, 8,...) followed by 0.
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LINKS
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FORMULA
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T(n,k) = A045545(k) if gcd(n,k) = 1, 0 otherwise, where A045545 = (1, 1, 2, 3, 7, 8, 22, 32, 66,...) starting with offset 1.
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EXAMPLE
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First few rows of the triangle =
1;
1, 0;
1, 1, 0;
1, 0, 2, 0;
1, 1, 2, 3, 0;
1, 0, 0, 0, 7, 0;
1, 1, 2, 3, 7, 8, 0;
1, 0, 2, 0, 7, 0, 22, 0;
1, 1, 0, 3, 7, 0, 22, 32, 0;
1, 0, 2, 0, 0, 0, 22, 0, 66, 0;
...
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MAPLE
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A045545:= n->`if`(n<3, 1, add(`if`(gcd(n, j)=1, A045545(j), 0), j=1..n-1) );
T:= (n, k) -> `if`(gcd(n, k)=1, A045545(k), 0);
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MATHEMATICA
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T[n_, k_]:= If[GCD[n, k]==1, A045545[k], 0];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Mar 08 2021 *)
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PROG
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(Sage)
@CachedFunction
def A045545(n): return 1 if n<3 else sum( kronecker_delta(gcd(n, j), 1)*A045545(j) for j in (0..n-1) )
def T(n, k): return A045545(k) if gcd(n, k)==1 else 0
flatten([[T(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 08 2021
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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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