A129312 A minimal 2 X 2 subdeterminant array.

1, 2, 2, 3, 5, 3, 4, 8, 8, 4, 5, 11, 13, 11, 5, 6, 14, 18, 18, 14, 6, 7, 17, 23, 25, 23, 17, 7, 8, 20, 28, 32, 32, 28, 20, 8, 9, 23, 33, 39, 41, 39, 33, 23, 9, 10, 26, 38, 46, 50, 50, 46, 38, 26, 10, 11, 29, 43, 53, 59, 61, 59, 53, 43, 29, 11, 12, 32, 48, 60, 68, 72, 72, 68, 60

Offset: 1

Author

Clark Kimberling, Apr 09 2007

Keywords

• nonn
• easy
• tabl

Comments

Given that row 1 and column 1 are the sequence (1,2,3,4,...), T is the array of minimal positive subdeterminants in the sense that for each 2 X 2 submatrix

a b

c d,

d is the least integer for which the resulting

determinant is positive; indeed, the determinant is 1.

T(n,n)=A001844(n).

SUM{T(n,k): k=1,2,...,n}=A081436(n).

When T is written as the triangle

1

2 2

3 5 3

4 8 8 4

5 11 13 11 5, etc.,

the row sums are A006527 and the alternating row sums are 1,0,1,0,1,0,1,0,... (A059841).

The underlying function T is the same as in A244418, but this triangle's rows hold n+k constant, while in A244418, n is held constant on each row, and k <= n.

T(n,k) can be interpreted as a figurate number, with an (n-1) x (k-1) rectangle of dots interleaved with an n x k rectangle. The American flag illustrates T(5,6).

Examples

			Northwest corner:
1 2 3 4 5 6
2 5 8 11 14 17
3 8 13 18 23 28
4 11 18 25 32 39
T(2,2)=5 because 5 is the least positive integer x for which the determinant of the 2 X 2 matrix below is positive:
1 2
2 x
		

Crossrefs

Cf. A001844, A081436, A006527, A059841.

Cf. A244418 (different triangle for the same function T).

Programs

Mathematica

(* Array version: *)
Grid[Table[SeriesCoefficient[Series[(n + (n - 1)*x)/(1 - x)^2, {x, 0, 12}], k], {n, 12}, {k, 0, 12}]] (* L. Edson Jeffery, Aug 23 2014 *)
(* Triangle version: *)
Grid[Table[SeriesCoefficient[Series[(n - k + (n - k - 1)*x)/(1 - x)^2, {x, 0, 12}], k], {n, 12}, {k, 0, n - 1}]] (* L. Edson Jeffery, Aug 23 2014 *)

Formula

T(n,k)=(2n-1)*k-n+1.

Extensions

Connection to A244418 and interpretation as figurate numbers from Allan C. Wechsler, Nov 18 2018