A125585 Array of constant-spaced integers read by antidiagonals.

1, 1, 2, 2, 3, 3, 1, 4, 5, 4, 2, 4, 6, 7, 5, 3, 5, 7, 8, 9, 6, 1, 6, 8, 10, 10, 11, 7, 2, 5, 9, 11, 13, 12, 13, 8, 3, 6, 9, 12, 14, 16, 14, 15, 9, 4, 7, 10, 13, 15, 17, 19, 16, 17, 10, 1, 8, 11, 14, 17, 18, 20, 22, 18, 19, 11, 2, 6, 12, 15, 18, 21, 21, 23, 25, 20, 21, 12

Offset

1,3

Comments

Iteratively taking sums of the values in each row starting with 1 produces the "figurate" numbers. For instance: 1, 1 + 2 = 3, 1 + 2 + 3 = 6 (the triangular numbers -- A000217) 1, 1 + 3 = 4, 1 + 3 + 5 = 9 (the square numbers -- A000290) 1, 1 + 4 = 5, 1 + 4 + 7 = 10 (the pentagonal numbers -- A000326) etc.

Iterative sums of the rows in between produce sequences related to the figurate numbers: 2, 2+4=6, 2+4+6=10 (oblong, or pronic, or heteromecic numbers -- A002378) 2, 2+5=7, 2+5+8=15 (second pentagonal numbers -- A005449) 3, 3+6=9, 3+6+9=18 (triangular matchstick numbers -- A045943) etc.

Iterative products produce the n-factorial numbers: 1, 1*3=3, 1*3*5=15 (double factorial numbers: (2*n-1)!! -- A001147) 2, 2*4=8, 2*4*6=48 (double factorial numbers: (2*n)!! -- A000165) 1, 1*4=4, 1*4*7=28, (triple factorial numbers (3*n-2)!!! -- A007559) etc.

Links

Alois P. Heinz, Antidiagonals n = 1..141

Example

The array begins:
  1, 2, 3,  4,  5,  6, ...
  1, 3, 5,  7,  9, 11, ...
  2, 4, 6,  8, 10, 12, ...
  1, 4, 7, 10, 13, 16, ...
  2, 5, 8, 11, 14, 17, ...
  3, 6, 9, 12, 15, 18, ...

Maple

A:= proc(n, k) local m;
      m:= floor((sqrt(8*n-7)-1)/2);
      n + (m+1)*(k-1-m/2)
    end:
seq(seq(A(1+d-k, k), k=1..d), d=1..12); # Alois P. Heinz, Jul 16 2012

Mathematica

imax = 5;
A = Table[k, {i, 1, imax}, {j, 1, i}, {k, j, j + i*imax*(imax+1)/2 - 1, i} ] // Flatten[#, 1]&;
Table[A[[n-k+1, k]], {n, 1, Length[A]}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 23 2016 *)

Crossrefs

Cf. A000217, A000290, A000326, A002378, A005449, A045943, A001147, A000165, A007559.

Columns k=1-2 give A002260, A108872.

Main diagonal gives A380548.

Sequence in context: A116464 A346136 A284532 * A327236 A191860 A109973

Adjacent sequences: A125582 A125583 A125584 * A125586 A125587 A125588

Keyword

easy,tabl,nonn

Author

Andrew S. Plewe, Jan 04 2007

Status

approved