A204008 Symmetric matrix based on f(i,j) = max{3i+j-3,i+3j-3}, by antidiagonals.

1, 4, 4, 7, 5, 7, 10, 8, 8, 10, 13, 11, 9, 11, 13, 16, 14, 12, 12, 14, 16, 19, 17, 15, 13, 15, 17, 19, 22, 20, 18, 16, 16, 18, 20, 22, 25, 23, 21, 19, 17, 19, 21, 23, 25, 28, 26, 24, 22, 20, 20, 22, 24, 26, 28, 31, 29, 27, 25, 23, 21, 23, 25, 27, 29, 31, 34, 32, 30

Offset: 1

Author

Clark Kimberling, Jan 09 2012

Keywords

• nonn
• tabl

Comments

A204008 represents the matrix M given by f(i,j)=max{3i+j-3,i+3j-3}for i>=1 and j>=1. See A204011 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

General case A206772. Let m be natural number. Table T(n,k)=max{m*n+k-m,n+m*k-m} read by antidiagonals.

For m=1 the result is A002024,

for m=2 the result is A204004,

for m=3 the result is A204008,

for m=4 the result is A206772. - Boris Putievskiy, Jan 24 2013

Examples

			Northwest corner:
   1,  4,  7, 10
   4,  5,  8, 11
   7,  8,  9, 12
  10, 11, 12, 13
		

Links

• Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732, 2012.

Crossrefs

Cf. A204011, A202453, A002024, A204004, A002260, A004736, A206772.

Programs

Mathematica

f[i_, j_] := Max[3 i + j - 3, 3 j + i - 3];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]   (* A204008 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]                (* A204011 *)
TableForm[Table[c[n], {n, 1, 10}]]

Formula

From Boris Putievskiy, Jan 24 2013: (Start)

For the general case, a(n) = m*A002024(n) + (m-1)*max{-A002260(n),-A004736(n)}.

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).

For m=3, a(n) = 3*(t+1) + 2*max{t*(t+1)/2-n,n-(t*t+3*t+4)/2}, where t=floor((-1+sqrt(8*n-7))/2). (End)