A391092 a(n) is the number of 2 X 2 matrices with elements 1..n where at least one row, at least one column and at least one of the two diagonals is strictly increasing.

0, 0, 2, 19, 76, 210, 470, 917, 1624, 2676, 4170, 6215, 8932, 12454, 16926, 22505, 29360, 37672, 47634, 59451, 73340, 89530, 108262, 129789, 154376, 182300, 213850, 249327, 289044, 333326, 382510, 436945, 496992, 563024, 635426, 714595, 800940, 894882, 996854, 1107301, 1226680, 1355460, 1494122

Offset

0,3

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = n*(n - 1)*(3*n^2 - 2*n - 2)/6.

G.f.: x^2*(2 + 9*x + x^2)/(1 - x)^5.

Mathematica

a[n_, d_] := a[n, d] = (AnyTrue[#, OrderedQ[#, Less] &] && AnyTrue[Transpose[#], OrderedQ[#, Less] &] && (OrderedQ[Diagonal[#], Less] || OrderedQ[Diagonal[Reverse[#, 2]], Less]) & /@ Tuples[Range[1, n], {d, d}]) // Boole // Total;
Table[a[n, 2], {n, 0, 42}]
A391092[n_] := n*(n^2*(3*n - 5) + 2)/6;
Array[A391092, 50, 0] (* Paolo Xausa, Dec 09 2025 *)

Crossrefs

Cf. A390922 (2 X 2, row and column), A390925 (3 X 3, row and column).

Cf. A206808 (2 X 2, row only), A390904 (3 X 3, row only).

Sequence in context: A219121 A054209 A256112 * A272053 A317274 A226019

Adjacent sequences: A391089 A391090 A391091 * A391093 A391094 A391095

Keyword

nonn,easy

Author

Robert P. P. McKone, Nov 28 2025

Status

approved