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%I #17 May 15 2016 11:39:30
%S 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,
%T 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,
%U 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
%N 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).
%C 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.
%F 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.
%e The table starts with rows of even length at n=2 as:
%e (1,1)
%e (empty)
%e (2,2)
%e (4,1),(1,4)
%e (3,3)
%e (5,2),(2,5)
%p A181633_row := proc(n)
%p local L,a,b;
%p L := [] ;
%p for a from n-1 to 1 by -1 do
%p b := n-a ;
%p if modp(a,3) = modp(b,3) then
%p L := [op(L),a,b] ;
%p end if;
%p end do:
%p L ;
%p end proc: # _R. J. Mathar_, May 14 2016
%t Table[Select[Transpose@{#, n - #}, Mod[First@ #, 3] == Mod[Last@ #, 3] &] &@ Reverse@ Range[1, n - 1], {n, 18}] // Flatten (* _Michael De Vlieger_, May 15 2016 *)
%Y Cf. A181634 (where q and q' may be zero), A008611 (half of the row lengths).
%K nonn,tabf,easy
%O 2,3
%A Florentin Smarandache (smarand(AT)unm.edu), Nov 03 2010
%E Edited by _R. J. Mathar_, May 14 2016
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