A129479 - OEIS

%I #11 Feb 12 2024 01:57:10

%S 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,

%T 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,

%U 4,2,1,1,0,0,0,0,1,1,12,0,0,0,0,0,0,0,0,0,0,1,1

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

%H G. C. Greubel, <a href="/A129479/b129479.txt">Rows n = 1..50 of the triangle, flattened</a>

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

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

%F From _G. C. Greubel_, Feb 11 2024: (Start)

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

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

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

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

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

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

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

%e First few rows of the triangle:

%e   1;

%e   2, 1;

%e   2, 1, 1;

%e   3, 1, 1, 1;

%e   4, 0, 0, 1, 1;

%e   4, 3, 1, 0, 1, 1;

%e   6, 0, 0, 0, 0, 1, 1;

%e   6, 2, 1, 1, 0, 0, 1, 1;

%e   ...

%t A054523[n_, k_]:= If[n==1, 1, If[Divisible[n,k], EulerPhi[n/k], 0]];

%t T[n_, k_]:= If[k<n, Sum[A054523[n, j+k], {j,0,1}], 1];

%t Table[T[n,k],{n,16},{k,n}]//Flatten (* _G. C. Greubel_, Feb 11 2024 *)

%o (Magma)

%o A054523:= func< n,k | n eq 1 select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;

%o A129479:= func< n,k | k le n-1 select A054523(n,k) + A054523(n,k+1) else 1 >;

%o [A129479(n,k): k in [1..n], n in [1..16]]; // _G. C. Greubel_, Feb 11 2024

%o (SageMath)

%o def A054523(n,k):

%o     if (k==n): return 1

%o     elif (n%k): return 0

%o     else: return euler_phi(n//k)

%o def A129479(n, k):

%o     if k<0 or k>n: return 0

%o     elif k==n: return 1

%o     else: return A054523(n,k) + A054523(n,k+1)

%o flatten([[A129479(n, k) for k in range(1,n+1)] for n in range(1,17)]) # _G. C. Greubel_, Feb 11 2024

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

%Y Cf. A000038, A019590, A054523, A054977.

%Y Cf. A063524, A097806, A126246, A183918.

%K nonn,tabl

%O 1,2

%A _Gary W. Adamson_, Apr 17 2007