A253902 Write numbers 1, then 2^2 down to 1, then 3^2 down to 1, then 4^2 down to 1 and so on.

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

Offset: 1

Author

Mikael Aaltonen, Jan 18 2015

Keywords

• nonn
• easy
• tabf

Comments

Triangle read by rows in which row n lists the first n^2 positive integers in decreasing order, n >= 1. - Omar E. Pol, Jan 20 2015

Examples

			From _Omar E. Pol_, Jan 20 2015: (Start)
Written as an irregular triangle in which row lengths are successive squares, the sequence begins:
   1;
   4,  3,  2,  1;
   9,  8,  7,  6,  5,  4,  3, 2, 1;
  16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1;
  ...
(End)
		

Crossrefs

Cf. A000290, A064866, A074279.

Programs

Maple

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

Mathematica

a253902[n_] := Flatten@ Table[Reverse[Range[i^2]], {i, n}]; a253902[6] (* Michael De Vlieger, Jan 19 2015 *)

PARI

lista(nn=10) = {for (n=1, nn, forstep(k=n^2, 1, -1, print1(k, ", ");););} \\ Michel Marcus, Jan 20 2015

Python

from sympy import integer_nthroot
def A253902(n): return (k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1))+2)*(k*((k<<1)-9)+13)//6-n # Chai Wah Wu, Nov 05 2024

Formula

For 1 <= n <= 650, a(n) = -n + (t + 2)*(2*t^2 - t + 3)/6, where t = floor((3*n)^(1/3)+1/2).

a(n) = k*(2*k^2-9*k+13)/6-n where k = floor((3*n)^(1/3))+3 if n>A000330(floor((3*n)^(1/3))) and k = floor((3*n)^(1/3))+2 otherwise. - Chai Wah Wu, Nov 05 2024