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

1,2

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

Links

Table of n, a(n) for n=1..91.

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

Example

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)

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 *)

Prog

(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

Crossrefs

Cf. A000290, A064866, A074279.

Sequence in context: A113778 A295673 A155172 * A349727 A129154 A055115

Adjacent sequences: A253899 A253900 A253901 * A253903 A253904 A253905

Keyword

nonn,easy,tabf

Author

Mikael Aaltonen, Jan 18 2015

Status

approved