A345018 For each n, append to the sequence n^2 consecutive integers, starting from n.

1, 2, 3, 4, 5, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41

Offset: 1

Author

Paolo Xausa, Jun 05 2021

Keywords

• nonn
• tabf

Comments

Irregular triangle read by rows T(n,k) in which row n lists the integers from n to n + n^2 - 1, with n >= 1.

Examples

			Written as an irregular triangle T(n,k) the sequence begins:
  n\k|  1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16 ...
  ---+---------------------------------------------------------------
   1 |  1;
   2 |  2,  3,  4,  5;
   3 |  3,  4,  5,  6,  7,  8,  9, 10, 11;
   4 |  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19;
  ...
		

Links

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

Crossrefs

Column 1: A000027.

Right border: A028387.

Row lengths: A000290.

Row sums: A255499.

Cf. A064866, A074279.

Programs

Maple

T:= n-> (t-> seq(n+i, i=0..t-1))(n^2):
seq(T(n), n=1..6);  # Alois P. Heinz, Nov 05 2024

Mathematica

Table[Range[n,n^2+n-1],{n,6}] (* Paolo Xausa, Sep 05 2023 *)

PARI

row(n) = vector(n^2, k, n+k-1); \\ Michel Marcus, Jun 08 2021

Python

from sympy import integer_nthroot
def A345018(n): return n-1+(k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))*(k*(3-(k<<1))+5)//6 # Chai Wah Wu, Nov 05 2024

Formula

T(n,k) = n + k - 1, with n >= 1 and 1 <= k <= n^2.