A053158 -id:A053158 - cp's OEIS frontend

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

Gary W. Adamson, Apr 17 2007

			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;
  ...

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

Cf. A000010 (alternating row sums), A053158 (row sums).
Cf. A000038, A019590, A054523, A054977.
Cf. A063524, A097806, A126246, A183918.

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

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 *)

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

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)

cp's OEIS Frontend

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

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