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A143656
Triangle T(n, k) = A045545(k) if gcd(n,k) = 1, 0 otherwise, read by rows.
2
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
OFFSET
1,9
COMMENTS
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.
Asymptotic limit of A054521^n * A143656 = A045545 as a vector.
FORMULA
Triangle read by rows, A054521 * (A045545 * 0^(n-k)); 1<=k<=n.
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.
EXAMPLE
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;
...
MAPLE
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);
seq(seq(T(n, k), k=1..n), n=1..12); # G. C. Greubel, Mar 08 2021
MATHEMATICA
A045545[n_]:= A045545[n]= If[n<3, 1, Sum[Boole[GCD[n, k]==1] A045545[k], {k, n-1}]];
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 *)
PROG
(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
CROSSREFS
Sequence in context: A257511 A039802 A126726 * A141169 A343887 A215075
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Aug 28 2008
STATUS
approved