A143656 Triangle T(n, k) = A045545(k) if gcd(n,k) = 1, 0 otherwise, read by rows.

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.

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

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

Cf. A045545, A054521.

Sequence in context: A257511 A039802 A126726 * A141169 A343887 A215075

Adjacent sequences: A143653 A143654 A143655 * A143657 A143658 A143659

Keyword

nonn,tabl

Author

Gary W. Adamson, Aug 28 2008

Status

approved