A204154 - OEIS

%I #29 Jan 07 2020 16:49:44

%S 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 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 18,15,12,9,6,9,12,15,18,21,23,20,17,14,11,8,8,11,14,17,20

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

%C 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 From _Nathaniel J. Strout_, Nov 14 2019: (Start)

%C The sum of the terms in the n-th "_|" shape is given by the octagonal numbers, A000567. For example,

%C       5,

%C       4,

%C   5,4,3,

%C   is considered the 3rd such shape.

%C The sum of the terms in the n-th antidiagonal is the absolute value of the (n+1)-th term of A266085. (End)

%H Robert Israel, <a href="/A204154/b204154.txt">Table of n, a(n) for n = 1..10011</a> (first 141 antidiagonals, flattened)

%F 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 Northwest corner:

%e   1, 3, 5, 7, 9, ...

%e   3, 2, 4, 6, 8, ...

%e   5, 4, 3, 5, 7, ...

%e   7, 6, 5, 4, 6, ...

%e   9, 8, 7, 6, 5, ...

%e   ...

%p seq(seq(max(3*j-m,2*m-3*j),j=1..m-1),m=2..19); # _Robert Israel_, Dec 03 2017

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

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

%t TableForm[m[8]] (* 8x8 principal submatrix *)

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

%t   {n, 1, 15}, {i, 1, n}]]  (* A204154 *)

%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[%]                 (* A204155 *)

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

%Y Cf. A204155, A204016, A202453.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jan 12 2012