A203990 Symmetric matrix based on f(i,j) = (i+j)*min(i,j), by antidiagonals.

2, 3, 3, 4, 8, 4, 5, 10, 10, 5, 6, 12, 18, 12, 6, 7, 14, 21, 21, 14, 7, 8, 16, 24, 32, 24, 16, 8, 9, 18, 27, 36, 36, 27, 18, 9, 10, 20, 30, 40, 50, 40, 30, 20, 10, 11, 22, 33, 44, 55, 55, 44, 33, 22, 11, 12, 24, 36, 48, 60, 72, 60, 48, 36, 24, 12, 13, 26, 39, 52, 65, 78, 78, 65, 52, 39, 26, 13

Offset: 1

Author

Clark Kimberling, Jan 09 2012

Keywords

• nonn
• tabl

Comments

This sequence represents the matrix M given by f(i,j) = (i+j)*min{i,j} for i >= 1 and j >= 1.

See A203991 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

Examples

			Northwest corner:
  2,  3,  4,  5,  6,  7
  3,  8, 10, 12, 14, 16
  4, 10, 18, 21, 24, 27
  5, 12, 21, 32, 36, 40
		

Links

• G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Crossrefs

Cf. A203991, A202453.

Programs

GAP

Flat(List([1..15], n-> List([1..n], k-> (n+1)*Minimum(n-k+1,k) ))); # G. C. Greubel, Jul 23 2019

Magma

[(n+1)*Min(n-k+1,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 23 2019

Mathematica

(* First program *)
f[i_, j_] := (i + j) Min[i, j];
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}]]  (* A203990 *)
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[%]              (* A203991 *)
TableForm[Table[c[n], {n, 1, 10}]]
(* Second program *)
Table[(n+1)*Min[n-k+1, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jul 23 2019 *)

PARI

for(n=1,15, for(k=1,n, print1((n+1)*min(n-k+1,k), ", "))) \\ G. C. Greubel, Jul 23 2019

Sage

[[(n+1)*min(n-k+1,k) for n in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 23 2019