A204022 - OEIS

%I #27 Sep 08 2022 08:46:01

%S 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,

%T 11,9,9,11,13,15,17,15,13,11,9,11,13,15,17,19,17,15,13,11,11,13,15,17,

%U 19,21,19,17,15,13,11,13,15,17,19,21,23,21,19,17,15,13,13,15,17,19,21,23

%N Symmetric matrix based on f(i,j) = max(2i-1, 2j-1), by antidiagonals.

%C 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.

%H G. C. Greubel, <a href="/A204022/b204022.txt">Rows n = 1..100 of triangle, flattened</a>

%F From _Ridouane Oudra_, May 27 2019: (Start)

%F a(n) = t + |t^2-2n+1|, where t = floor(sqrt(2n-1)+1/2).

%F a(n) = A209302(2n-1).

%F a(n) = A002024(n) + |A002024(n)^2-2n+1|.

%F a(n) = t + |t^2-2n+1|, where t = floor(sqrt(2n)+1/2). (End)

%e Northwest corner:

%e   1 3 5 7 9

%e   3 3 5 7 9

%e   5 5 5 7 9

%e   7 7 7 7 9

%e   9 9 9 9 9

%t (* First program *)

%t f[i_, j_] := Max[2 i - 1, 2 j - 1];

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

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

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

%t   {n, 15}, {i, n}]]                (* A204022 *)

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

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

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

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

%t Flatten[%]                         (* A204023 *)

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

%t (* Second program *)

%t Table[Max[2*k-1, 2*(n-k)+1], {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Jul 23 2019 *)

%o (PARI) {T(n, k) = max(2*k-1, 2*(n-k)+1)};

%o for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ _G. C. Greubel_, Jul 23 2019

%o (Magma) [[Max(2*k-1, 2*(n-k)+1): k in [1..n]]: n in [1..12]]; // _G. C. Greubel_, Jul 23 2019

%o (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

%o (GAP) Flat(List([1..12], n-> List([1..n], k-> Maximum(2*k-1, 2*(n-k)+1) ))); # _G. C. Greubel_, Jul 23 2019

%Y Cf. A202453, A204016, A204022, A209302.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jan 11 2012