A386431 Triangle T(n,k) read by rows, where row n contains the numbers 1, ..., n^2, written columnwise into a square matrix and read rowwise.

1, 1, 3, 2, 4, 1, 4, 7, 2, 5, 8, 3, 6, 9, 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 15, 4, 8, 12, 16, 1, 6, 11, 16, 21, 2, 7, 12, 17, 22, 3, 8, 13, 18, 23, 4, 9, 14, 19, 24, 5, 10, 15, 20, 25, 1, 7, 13, 19, 25, 31, 2, 8, 14, 20, 26, 32, 3, 9, 15, 21, 27, 33, 4, 10, 16, 22, 28, 34, 5, 11, 17, 23, 29, 35, 6, 12, 18, 24, 30, 36

Offset

1,3

Links

Paolo Xausa, Table of n, a(n) for n = 1..10416 (rows 1..31 of triangle, flattened).

Formula

T(n,k) = 1 + n*k - n - (n^2-1)*floor((k-1)/n). - Alois P. Heinz, Sep 02 2025

Example

First four rows:
 n = 1:
  1.
 n = 2:
  1, 3,
  2, 4.
 n = 3:
  1, 4, 7,
  2, 5, 8,
  3, 6, 9.
 n = 4:
  1, 5, 9,  13,
  2, 6, 10, 14,
  3, 7, 11, 15,
  4, 8, 12, 16.

Maple

T:= (n, k)-> 1+n*k-n-(n^2-1)*floor((k-1)/n):
seq(seq(T(n, k), k=1..n^2), n=1..7);  # Alois P. Heinz, Sep 02 2025

Mathematica

A386431[n_, k_] := 1 + n*k - n - (n^2 - 1)*Quotient[k - 1, n];
Table[A386431[n, k], {n, 6}, {k, n^2}] (* or *)
A386431row[n_] := Flatten[Transpose[Partition[Range[n^2], n]]];
Array[A386431row, 6] (* Paolo Xausa, Sep 09 2025 *)

Prog

(PARI) a30(n) = n*(n+1)*(2*n+1) / 6;
f(k) = my(n=1); while(a30(n-1) < k, n++); n-1; \\ A074279
a(k) = my(n=f(k)); ((k-a30(n-1)-1) % n) * n + floor((k-a30(n-1)-1)/n)+1 ; \\ Michel Marcus, Sep 02 2025

Crossrefs

Cf. A000330, A064866, A074279.

Row sums give A037270.

T(n,n) gives A002061.

Sequence in context: A187760 A122143 A144868 * A134029 A117623 A145690

Adjacent sequences: A386428 A386429 A386430 * A386432 A386433 A386434

Keyword

nonn,easy,tabf

Author

Binay Krishna Maity, Aug 25 2025

Status

approved