A380548 -id:A380548 - cp's OEIS frontend

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

Andrew S. Plewe, Jan 04 2007

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.

			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, ...

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

Cf. A000217, A000290, A000326, A002378, A005449, A045943, A001147, A000165, A007559.
Columns k=1-2 give A002260, A108872.
Main diagonal gives A380548.

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

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 *)

cp's OEIS Frontend

A125585 Array of constant-spaced integers read by antidiagonals.

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