

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, 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
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OFFSET

2,3


COMMENTS

If (s,t) is a pair in the sequence, then (s+3u,t3u) is also a pair in the sequence for any integer u such that both s+3u > 0 and t3u > 0.


LINKS



FORMULA

i) If n is even, n=2k, then its pairs are: (k+3p,k3p), where p is an integer such that both k+3p > 0 and k3p > 0. ii) If n is odd, n=2k+1, then its pairs are (k+3p+2,k3p1), where p is an integer such that both k+3p+2 > 0 and k3p1 > 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

local L, a, b;
L := [] ;
for a from n1 to 1 by 1 do
b := na ;
if modp(a, 3) = modp(b, 3) then
L := [op(L), a, b] ;
end if;
end do:
L ;


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).


KEYWORD

nonn,tabf,easy


AUTHOR

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


EXTENSIONS



STATUS

approved



