This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A129312 #29 May 26 2025 19:25:05
%S A129312 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 A129312 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 A129312 38,26,10,11,29,43,53,59,61,59,53,43,29,11,12,32,48,60,68,72,72,68,60
%N A129312 A minimal 2 X 2 subdeterminant array.
%C A129312 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 A129312 a b
%C A129312 c d,
%C A129312 d is the least integer for which the resulting
%C A129312 determinant is positive; indeed, the determinant is 1.
%C A129312 T(n,n)=A001844(n).
%C A129312 SUM{T(n,k): k=1,2,...,n}=A081436(n).
%C A129312 When T is written as the triangle
%C A129312 1
%C A129312 2 2
%C A129312 3 5 3
%C A129312 4 8 8 4
%C A129312 5 11 13 11 5, etc.,
%C A129312 the row sums are A006527 and the alternating row sums are 1,0,1,0,1,0,1,0,... (A059841).
%C A129312 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 A129312 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 A129312 T(n,k)=(2n-1)*k-n+1.
%e A129312 Northwest corner:
%e A129312 1 2 3 4 5 6
%e A129312 2 5 8 11 14 17
%e A129312 3 8 13 18 23 28
%e A129312 4 11 18 25 32 39
%e A129312 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 A129312 1 2
%e A129312 2 x
%t A129312 (* Array version: *)
%t A129312 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 A129312 (* Triangle version: *)
%t A129312 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 A129312 Cf. A001844, A081436, A006527, A059841.
%Y A129312 Cf. A244418 (different triangle for the same function T).
%K A129312 nonn,easy,tabl
%O A129312 1,2
%A A129312 _Clark Kimberling_, Apr 09 2007
%E A129312 Connection to A244418 and interpretation as figurate numbers from _Allan C. Wechsler_, Nov 18 2018