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A minimal 2 X 2 subdeterminant array.
1

%I #14 Nov 18 2018 09:49:42

%S 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,

%T 8,20,28,32,32,28,20,8,9,23,33,39,41,39,33,23,9,10,26,38,46,50,50,46,

%U 38,26,10,11,29,43,53,59,61,59,53,43,29,11,12,32,48,60,68,72,72,68,60

%N A minimal 2 X 2 subdeterminant array.

%C 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

%C a b

%C c d,

%C d is the least integer for which the resulting

%C determinant is positive; indeed, the determinant is 1.

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

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

%C When T is written as the triangle

%C 1

%C 2 2

%C 3 5 3

%C 4 8 8 4

%C 5 11 13 11 5, etc.,

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

%C 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.

%C 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).

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

%e Northwest corner:

%e 1 2 3 4 5 6

%e 2 5 8 11 14 17

%e 3 8 13 18 23 28

%e 4 11 18 25 32 39

%e 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:

%e 1 2

%e 2 x

%t (* Array version: *)

%t 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 *)

%t (* Triangle version: *)

%t 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 *)

%Y Cf. A001844, A081436, A006527, A059841.

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

%K nonn,easy,tabl

%O 1,2

%A _Clark Kimberling_, Apr 09 2007

%E Connection to A244418 and interpretation as figurate numbers from _Allan C. Wechsler_, Nov 18 2018