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

1,2

Links

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

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)

Example

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;
  ...

Mathematica

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

Prog

(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
(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

Crossrefs

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

Cf. A000038, A019590, A054523, A054977.

Cf. A063524, A097806, A126246, A183918.

Sequence in context: A084610 A303336 A361462 * A261095 A249809 A075104

Adjacent sequences: A129476 A129477 A129478 * A129480 A129481 A129482

Keyword

nonn,tabl

Author

Gary W. Adamson, Apr 17 2007

Status

approved