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Symmetric matrix based on f(i,j) = (i+j)*min{i,j}, by antidiagonals.
3

%I #13 Sep 08 2022 08:46:01

%S 1,0,0,-1,1,-1,-2,0,0,-2,-3,-1,1,-1,-3,-4,-2,0,0,-2,-4,-5,-3,-1,1,-1,

%T -3,-5,-6,-4,-2,0,0,-2,-4,-6,-7,-5,-3,-1,1,-1,-3,-5,-7,-8,-6,-4,-2,0,

%U 0,-2,-4,-6,-8,-9,-7,-5,-3,-1,1,-1,-3,-5,-7,-9

%N Symmetric matrix based on f(i,j) = (i+j)*min{i,j}, by antidiagonals.

%C A203994 represents the matrix M given by f(i,j) = min(i-j+1,j-i+1) for i >= 1 and j >= 1. See A203995 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

%H G. C. Greubel, <a href="/A203994/b203994.txt">Antidiagonal rows n = 1..100, flattened</a>

%e Northwest corner:

%e 1 0 -1 -2 -3

%e 0 1 0 -1 -2

%e -1 0 1 0 -1

%e 2 -1 0 1 0

%t (* First program *)

%t f[i_, j_] := Min[i - j + 1, j - i + 1];

%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

%t TableForm[m[6]] (* 6 X 6 principal submatrix *)

%t Flatten[Table[f[i, n + 1 - i],

%t {n, 1, 12}, {i, 1, n}]] (* A203994 *)

%t p[n_] := CharacteristicPolynomial[m[n], x];

%t c[n_] := CoefficientList[p[n], x]

%t TableForm[Flatten[Table[p[n], {n, 1, 10}]]]

%t Table[c[n], {n, 1, 12}]

%t Flatten[%] (* A203995 *)

%t TableForm[Table[c[n], {n, 1, 10}]]

%t (* Second program *)

%t Table[Min[2*k-n, n-2*k+2], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Jul 23 2019 *)

%o (PARI) for(n=1,15, for(k=1,n, print1(min(2*k-n, n-2*k+2), ", "))) \\ _G. C. Greubel_, Jul 23 2019

%o (Magma) [Min(2*k-n, n-2*k+2): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jul 23 2019

%o (Sage) [[min(2*k-n, n-2*k+2) for k in (1..n)] for n in (1..15)] # _G. C. Greubel_, Jul 23 2019

%o (GAP) Flat(List([1..15], n-> List([1..n], k-> Minimum(2*k-n, n-2*k+2) ))); # _G. C. Greubel_, Jul 23 2019

%Y Cf. A203995, A202453.

%K tabl,sign

%O 1,7

%A _Clark Kimberling_, Jan 09 2012