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

Author

Gary W. Adamson, Aug 28 2008

Keywords

• nonn
• tabl

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.

Examples

			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;
  ...
		

Links

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

Crossrefs

Cf. A045545, A054521.

Programs

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 *)

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

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.