A181633 Irregular triangle read by rows, where row n contains the pairs [q,q'] of all compositions n=q+q' with q,q'>0 and q == q' (mod 3).

1, 1, 2, 2, 4, 1, 1, 4, 3, 3, 5, 2, 2, 5, 7, 1, 4, 4, 1, 7, 6, 3, 3, 6, 8, 2, 5, 5, 2, 8, 10, 1, 7, 4, 4, 7, 1, 10, 9, 3, 6, 6, 3, 9, 11, 2, 8, 5, 5, 8, 2, 11, 13, 1, 10, 4, 7, 7, 4, 10, 1, 13, 12, 3, 9, 6, 6, 9, 3, 12, 14, 2, 11, 5, 8, 8, 5, 11, 2, 14, 16, 1, 13, 4, 10, 7, 7, 10, 4, 13, 1, 16, 15, 3, 12, 6, 9, 9, 6, 12, 3, 15

Offset: 2

Author

Florentin Smarandache (smarand(AT)unm.edu), Nov 03 2010

Keywords

• nonn
• tabf
• easy

Comments

If (s,t) is a pair in the sequence, then (s+3u,t-3u) is also a pair in the sequence for any integer u such that both s+3u > 0 and t-3u > 0.

Examples

			The table starts with rows of even length at n=2 as:
(1,1)
(empty)
(2,2)
(4,1),(1,4)
(3,3)
(5,2),(2,5)
		

Crossrefs

Cf. A181634 (where q and q' may be zero), A008611 (half of the row lengths).

Programs

Maple

A181633_row := proc(n)
    local L,a,b;
    L := [] ;
    for a from n-1 to 1 by -1 do
        b := n-a ;
        if modp(a,3) = modp(b,3) then
            L := [op(L),a,b] ;
        end if;
    end do:
    L ;
end proc: # R. J. Mathar, May 14 2016

Mathematica

Table[Select[Transpose@{#, n - #}, Mod[First@ #, 3] == Mod[Last@ #, 3] &] &@ Reverse@ Range[1, n - 1], {n, 18}] // Flatten (* Michael De Vlieger, May 15 2016 *)

Formula

i) If n is even, n=2k, then its pairs are: (k+3p,k-3p), where p is an integer such that both k+3p > 0 and k-3p > 0. ii) If n is odd, n=2k+1, then its pairs are (k+3p+2,k-3p-1), where p is an integer such that both k+3p+2 > 0 and k-3p-1 > 0.

Extensions

Edited by R. J. Mathar, May 14 2016