cp's OEIS Frontend

A204004 Symmetric matrix based on f(i,j) = max{2i+j-2,i+2j-2}, by antidiagonals.

%I A204004 #23 Jun 24 2017 00:58:52
%S A204004 1,3,3,5,4,5,7,6,6,7,9,8,7,8,9,11,10,9,9,10,11,13,12,11,10,11,12,13,
%T A204004 15,14,13,12,12,13,14,15,17,16,15,14,13,14,15,16,17,19,18,17,16,15,15,
%U A204004 16,17,18,19,21,20,19,18,17,16,17,18,19,20,21,23,22,21,20,19,18
%N A204004 Symmetric matrix based on f(i,j) = max{2i+j-2,i+2j-2}, by antidiagonals.
%C A204004 A204004 represents the matrix M given by f(i,j)=max{2i+j,i+2j}for i>=1 and j>=1. See A204005 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
%C A204004 General case A206772. Let m be natural number. Table T(n,k)=max{m*n+k-m,n+m*k-m} read by antidiagonals.
%C A204004   For m=1 the result is A002024,
%C A204004   for m=2 the result is A204004,
%C A204004   for m=3 the result is A204008,
%C A204004   for m=4 the result is A206772. - _Boris Putievskiy_, Jan 24 2013
%H A204004 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [Of] Integer Sequences And Pairing Functions</a>, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
%F A204004 From _Boris Putievskiy_, Jan 24 2013: (Start)
%F A204004 For the general case, a(n) = m*A002024(n) + (m-1)*max{-A002260(n),-A004736(n)}.
%F A204004 a(n) = m*(t+1) + (m-1)*max{t*(t+1)/2-n,n-(t*t+3*t+4)/2}, where t=floor((-1+sqrt(8*n-7))/2).
%F A204004 For m=2, a(n) = 2*(t+1) + max{t*(t+1)/2-n,n-(t*t+3*t+4)/2}, where t=floor((-1+sqrt(8*n-7))/2). (End)
%e A204004 Northwest corner:
%e A204004   1,  3,  5,  7,  9
%e A204004   3,  4,  6,  8, 10
%e A204004   5,  6,  7,  9, 11
%e A204004   7,  8,  9, 10, 12
%t A204004 f[i_, j_] := Max[2 i + j - 2, 2 j + i - 2];
%t A204004 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204004 TableForm[m[6]] (* 6x6 principal submatrix *)
%t A204004 Flatten[Table[f[i, n + 1 - i],
%t A204004 {n, 1, 12}, {i, 1, n}]]  (* A204004 *)
%t A204004 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204004 c[n_] := CoefficientList[p[n], x]
%t A204004 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204004 Table[c[n], {n, 1, 12}]
%t A204004 Flatten[%]   (* A204005 *)
%t A204004 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204004 Cf. A204005, A202453, A002024, A204008, A002260, A004736, A206772.
%K A204004 nonn,tabl
%O A204004 1,2
%A A204004 _Clark Kimberling_, Jan 09 2012