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A204154 Symmetric matrix based on f(i,j) = max(2i-j, 2j-i), by antidiagonals.

%I A204154 #29 Jan 07 2020 16:49:44
%S A204154 1,3,3,5,2,5,7,4,4,7,9,6,3,6,9,11,8,5,5,8,11,13,10,7,4,7,10,13,15,12,
%T A204154 9,6,6,9,12,15,17,14,11,8,5,8,11,14,17,19,16,13,10,7,7,10,13,16,19,21,
%U A204154 18,15,12,9,6,9,12,15,18,21,23,20,17,14,11,8,8,11,14,17,20
%N A204154 Symmetric matrix based on f(i,j) = max(2i-j, 2j-i), by antidiagonals.
%C A204154 A204154 represents the matrix M given by f(i,j) = max(2i-j, 2j-i) for i >= 1 and j >= 1.  See A204155 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.
%C A204154 From _Nathaniel J. Strout_, Nov 14 2019: (Start)
%C A204154 The sum of the terms in the n-th "_|" shape is given by the octagonal numbers, A000567. For example,
%C A204154       5,
%C A204154       4,
%C A204154   5,4,3,
%C A204154   is considered the 3rd such shape.
%C A204154 The sum of the terms in the n-th antidiagonal is the absolute value of the (n+1)-th term of A266085. (End)
%H A204154 Robert Israel, <a href="/A204154/b204154.txt">Table of n, a(n) for n = 1..10011</a> (first 141 antidiagonals, flattened)
%F A204154 G.f. as array: (1 + x + y - 7*y*x + 2*y*x^2 + 2*y^2*x)*x*y/((1-x*y)*(1-x)^2*(1-y)^2). - _Robert Israel_, Dec 03 2017
%e A204154 Northwest corner:
%e A204154   1, 3, 5, 7, 9, ...
%e A204154   3, 2, 4, 6, 8, ...
%e A204154   5, 4, 3, 5, 7, ...
%e A204154   7, 6, 5, 4, 6, ...
%e A204154   9, 8, 7, 6, 5, ...
%e A204154   ...
%p A204154 seq(seq(max(3*j-m,2*m-3*j),j=1..m-1),m=2..19); # _Robert Israel_, Dec 03 2017
%t A204154 f[i_, j_] := Max[2 i - j, 2 j - i];
%t A204154 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204154 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204154 Flatten[Table[f[i, n + 1 - i],
%t A204154   {n, 1, 15}, {i, 1, n}]]  (* A204154 *)
%t A204154 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204154 c[n_] := CoefficientList[p[n], x]
%t A204154 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204154 Table[c[n], {n, 1, 12}]
%t A204154 Flatten[%]                 (* A204155 *)
%t A204154 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204154 Cf. A204155, A204016, A202453.
%K A204154 nonn,tabl
%O A204154 1,2
%A A204154 _Clark Kimberling_, Jan 12 2012