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A368197
Triangle read by rows: T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z), n) = k], where f(x,y,z) = x^2 + y^2 - z^2.
2
1, 4, 4, 18, 0, 9, 32, 8, 0, 24, 100, 0, 0, 0, 25, 72, 72, 36, 0, 0, 36, 294, 0, 0, 0, 0, 0, 49, 256, 64, 0, 96, 0, 0, 0, 96, 486, 0, 144, 0, 0, 0, 0, 0, 99, 400, 400, 0, 0, 100, 0, 0, 0, 0, 100, 1210, 0, 0, 0, 0, 0, 0, 0, 0, 0, 121
OFFSET
1,2
COMMENTS
Row n has sum n^3. The number of nonzero terms in row n appears to be A000005(n). It appears that Sum_{k=1..n} T(n,k)*A023900(k) = A063524(n). Main diagonal appears to be A062775. First column appears to be A053191.
It appears that when p > 2 in f(x,y,z,p) = x^p + y^p - z^p and T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z,p), n) = k], then Sum_{k=1..n} T(n,k)*A023900(k) is not equal to A063524(n). - Mats Granvik, May 07 2024
FORMULA
T(n,k) = Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z), n) = k], where f(x,y,z) = x^2 + y^2 - z^2.
EXAMPLE
Triangle begins:
1;
4, 4;
18, 0, 9;
32, 8, 0, 24;
100, 0, 0, 0, 25;
72, 72, 36, 0, 0, 36;
294, 0, 0, 0, 0, 0, 49;
256, 64, 0, 96, 0, 0, 0, 96;
486, 0, 144, 0, 0, 0, 0, 0, 99;
400, 400, 0, 0, 100, 0, 0, 0, 0, 100;
1210, 0, 0, 0, 0, 0, 0, 0, 0, 0, 121;
...
MATHEMATICA
nn = 11; p = 2; f = x^p + y^p - z^p; Flatten[Table[Table[Sum[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}], {z, 1, n}], {k, 1, n}], {n, 1, nn}]]
KEYWORD
nonn,tabl
AUTHOR
Mats Granvik, Dec 16 2023
STATUS
approved