cp's OEIS Frontend

A125585 Array of constant-spaced integers read by antidiagonals.

%I A125585 #24 Jan 26 2025 16:25:12
%S A125585 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,
%T A125585 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,
%U A125585 14,17,18,20,22,18,19,11,2,6,12,15,18,21,21,23,25,20,21,12
%N A125585 Array of constant-spaced integers read by antidiagonals.
%C A125585 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.
%C A125585 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.
%C A125585 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.
%H A125585 Alois P. Heinz, <a href="/A125585/b125585.txt">Antidiagonals n = 1..141</a>
%e A125585 The array begins:
%e A125585   1, 2, 3,  4,  5,  6, ...
%e A125585   1, 3, 5,  7,  9, 11, ...
%e A125585   2, 4, 6,  8, 10, 12, ...
%e A125585   1, 4, 7, 10, 13, 16, ...
%e A125585   2, 5, 8, 11, 14, 17, ...
%e A125585   3, 6, 9, 12, 15, 18, ...
%p A125585 A:= proc(n, k) local m;
%p A125585       m:= floor((sqrt(8*n-7)-1)/2);
%p A125585       n + (m+1)*(k-1-m/2)
%p A125585     end:
%p A125585 seq(seq(A(1+d-k, k), k=1..d), d=1..12); # _Alois P. Heinz_, Jul 16 2012
%t A125585 imax = 5;
%t A125585 A = Table[k, {i, 1, imax}, {j, 1, i}, {k, j, j + i*imax*(imax+1)/2 - 1, i} ] // Flatten[#, 1]&;
%t A125585 Table[A[[n-k+1, k]], {n, 1, Length[A]}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, May 23 2016 *)
%Y A125585 Cf. A000217, A000290, A000326, A002378, A005449, A045943, A001147, A000165, A007559.
%Y A125585 Columns k=1-2 give A002260, A108872.
%Y A125585 Main diagonal gives A380548.
%K A125585 easy,tabl,nonn
%O A125585 1,3
%A A125585 _Andrew S. Plewe_, Jan 04 2007