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A204022
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Symmetric matrix based on f(i,j) = max(2i-1, 2j-1), by antidiagonals.
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4
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1, 3, 3, 5, 3, 5, 7, 5, 5, 7, 9, 7, 5, 7, 9, 11, 9, 7, 7, 9, 11, 13, 11, 9, 7, 9, 11, 13, 15, 13, 11, 9, 9, 11, 13, 15, 17, 15, 13, 11, 9, 11, 13, 15, 17, 19, 17, 15, 13, 11, 11, 13, 15, 17, 19, 21, 19, 17, 15, 13, 11, 13, 15, 17, 19, 21, 23, 21, 19, 17, 15, 13, 13, 15, 17, 19, 21, 23
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OFFSET
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1,2
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COMMENTS
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This sequence represents the matrix M given by f(i,j) = max(2i-1, 2j-1) for i >= 1 and j >= 1. See A204023 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.
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LINKS
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FORMULA
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a(n) = t + |t^2-2n+1|, where t = floor(sqrt(2n-1)+1/2).
a(n) = t + |t^2-2n+1|, where t = floor(sqrt(2n)+1/2). (End)
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EXAMPLE
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Northwest corner:
1 3 5 7 9
3 3 5 7 9
5 5 5 7 9
7 7 7 7 9
9 9 9 9 9
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MATHEMATICA
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(* First program *)
f[i_, j_] := Max[2 i - 1, 2 j - 1];
m[n_] := Table[f[i, j], {i, n}, {j, n}]
TableForm[m[6]] (* 6 X 6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 10}]]]
Table[c[n], {n, 12}]
TableForm[Table[c[n], {n, 10}]]
(* Second program *)
Table[Max[2*k-1, 2*(n-k)+1], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 23 2019 *)
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PROG
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(PARI) {T(n, k) = max(2*k-1, 2*(n-k)+1)};
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 23 2019
(Magma) [[Max(2*k-1, 2*(n-k)+1): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 23 2019
(Sage) [[max(2*k-1, 2*(n-k)+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 23 2019
(GAP) Flat(List([1..12], n-> List([1..n], k-> Maximum(2*k-1, 2*(n-k)+1) ))); # G. C. Greubel, Jul 23 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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