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

1,2

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).

Links

Table of n, a(n) for n=1..75.

Formula

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

Example

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

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 *)

Crossrefs

Cf. A001844, A081436, A006527, A059841.

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

Sequence in context: A132071 A334923 A061177 * A115262 A128141 A252829

Adjacent sequences: A129309 A129310 A129311 * A129313 A129314 A129315

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Apr 09 2007

Extensions

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

Status

approved