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%I #5 Mar 08 2021 22:26:15
%S 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,
%T 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,
%U 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
%N Triangle T(n, k) = A045545(k) if gcd(n,k) = 1, 0 otherwise, read by rows.
%C Sum of row terms = A045545 starting with offset 1: (1, 1, 2, 3, 7, 8, 22,...).
%C A045545 also = rightmost diagonal with nonzero terms.
%C Sum of n-th row terms = rightmost nonzero term of next row.
%C Prime n rows = first (n-1) terms of (1, 1, 2, 3, 7, 8,...) followed by 0.
%C Asymptotic limit of A054521^n * A143656 = A045545 as a vector.
%H G. C. Greubel, <a href="/A143656/b143656.txt">Rows n = 1..50 of the triangle, flattened</a>
%F Triangle read by rows, A054521 * (A045545 * 0^(n-k)); 1<=k<=n.
%F 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.
%e First few rows of the triangle =
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 1, 0, 2, 0;
%e 1, 1, 2, 3, 0;
%e 1, 0, 0, 0, 7, 0;
%e 1, 1, 2, 3, 7, 8, 0;
%e 1, 0, 2, 0, 7, 0, 22, 0;
%e 1, 1, 0, 3, 7, 0, 22, 32, 0;
%e 1, 0, 2, 0, 0, 0, 22, 0, 66, 0;
%e ...
%p A045545:= n->`if`(n<3, 1, add(`if`(gcd(n,j)=1, A045545(j), 0), j=1..n-1) );
%p T:= (n,k) -> `if`(gcd(n,k)=1, A045545(k), 0);
%p seq(seq(T(n,k), k=1..n), n=1..12); # _G. C. Greubel_, Mar 08 2021
%t A045545[n_]:= A045545[n]= If[n<3, 1, Sum[Boole[GCD[n, k]==1] A045545[k], {k,n-1}]];
%t T[n_, k_]:= If[GCD[n, k]==1, A045545[k], 0];
%t Table[T[n, k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 08 2021 *)
%o (Sage)
%o @CachedFunction
%o def A045545(n): return 1 if n<3 else sum( kronecker_delta(gcd(n, j), 1)*A045545(j) for j in (0..n-1) )
%o def T(n,k): return A045545(k) if gcd(n,k)==1 else 0
%o flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 08 2021
%Y Cf. A045545, A054521.
%K nonn,tabl
%O 1,9
%A _Gary W. Adamson_, Aug 28 2008
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