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A125585 Array of constant-spaced integers read by antidiagonals. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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: (2n-1)!! -- A001147 2, 2*4=8, 2*4*6=48 (Double factorial numbers: (2n)!! -- 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-Fran├žois Alcover, May 23 2016 *)

CROSSREFS

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

Sequence in context: A242361 A116464 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

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Last modified April 22 16:32 EDT 2021. Contains 343177 sequences. (Running on oeis4.)