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

2,3

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.

Links

Table of n, a(n) for n=2..103.

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.

Example

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)

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 *)

Crossrefs

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

Sequence in context: A214741 A243487 A143485 * A245204 A099320 A206714

Adjacent sequences: A181630 A181631 A181632 * A181634 A181635 A181636

Keyword

nonn,tabf,easy

Author

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

Extensions

Edited by R. J. Mathar, May 14 2016

Status

approved