A129479 Triangle read by rows: A054523 * A097806 as infinite lower triangular matrices.

1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 4, 0, 0, 1, 1, 4, 3, 1, 0, 1, 1, 6, 0, 0, 0, 0, 1, 1, 6, 2, 1, 1, 0, 0, 1, 1, 6, 2, 2, 0, 0, 0, 0, 1, 1, 8, 4, 0, 1, 1, 0, 0, 0, 1, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 4, 4, 2, 1, 1, 0, 0, 0, 0, 1, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Author

Gary W. Adamson, Apr 17 2007

Keywords

• nonn
• tabl

Examples

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  3, 1, 1, 1;
  4, 0, 0, 1, 1;
  4, 3, 1, 0, 1, 1;
  6, 0, 0, 0, 0, 1, 1;
  6, 2, 1, 1, 0, 0, 1, 1;
  ...
		

Links

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

Crossrefs

Cf. A000010 (alternating row sums), A053158 (row sums).

Cf. A000038, A019590, A054523, A054977.

Cf. A063524, A097806, A126246, A183918.

Programs

Magma

A054523:= func< n,k | n eq 1 select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;
A129479:= func< n,k | k le n-1 select A054523(n,k) + A054523(n,k+1) else 1 >;
[A129479(n,k): k in [1..n], n in [1..16]]; // G. C. Greubel, Feb 11 2024

Mathematica

A054523[n_, k_]:= If[n==1, 1, If[Divisible[n,k], EulerPhi[n/k], 0]];
T[n_, k_]:= If[kA054523[n, j+k], {j,0,1}], 1];
Table[T[n,k],{n,16},{k,n}]//Flatten (* G. C. Greubel, Feb 11 2024 *)

SageMath

def A054523(n,k):
    if (k==n): return 1
    elif (n%k): return 0
    else: return euler_phi(n//k)
def A129479(n, k):
    if k<0 or k>n: return 0
    elif k==n: return 1
    else: return A054523(n,k) + A054523(n,k+1)
flatten([[A129479(n, k) for k in range(1,n+1)] for n in range(1,17)]) # G. C. Greubel, Feb 11 2024

Formula

Sum_{k=1..n} T(n, k) = A053158(n) (row sums).

T(n, 1) = A126246(n).

From G. C. Greubel, Feb 11 2024: (Start)

T(n, k) = A054523(n, k) + A054523(n, k+1) for k < n, otherwise 1.

T(2*n-1, n) = A019590(n).

T(2*n, n) = A054977(n).

T(2*n+1, n) = A000038(n).

T(3*n, n) = A063524(n-1).

T(3*n-2, n) = A183918(n+2).

Sum_{k=1..n} (-1)^(k-1) * T(n, k) = A000010(n). (End)