%I #2 Mar 30 2012 17:34:39
%S 1,1,2,1,3,4,1,4,6,7,1,5,7,9,11,1,6,9,12,14,16,1,7,10,13,16,19,21,1,8,
%T 12,15,18,21,24,27,1,9,13,17,21,24,27,30,33,1,10,15,20,25,28,32,35,39,
%U 42,1,11,16,21,26,30,34,38,42,46,50,1,12,18,23,28,33,38,42,46,50,54,58
%N An averaging sum sequence based on( improved):a(n,m,q)=Floor[(a(n - 1, m - 1,q) + a(n - 1, m,q))/2] with limit q
%C Row sums are:
%C {1, 3, 8, 18, 33, 58, 87, 126, 175, 247, 315, 403, 510,...}.
%C First Mathematica version of this sequence didn't average right for q levels.
%C Example a(n,m) for q=10:
%C {1},
%C {1, 10},
%C {1, 5, 10},
%C {1, 3, 7, 10},
%C {1, 2, 5, 8, 10},
%C {1, 1, 3, 6, 9, 10},
%C {1, 1, 2, 4, 7, 9, 10},
%C {1, 1, 1, 3, 5, 8, 9, 10},
%C {1, 1, 1, 2, 4, 6, 8, 9, 10},
%C {1, 1, 1, 1, 3, 5, 7, 8, 9, 10},
%C {1, 1, 1, 1, 2, 4, 6, 7, 8, 9, 10}
%F a(n,m,q)=Floor[(a(n - 1, m - 1,q) + a(n - 1, m,q))/2] with limit q:
%F t(n,q)=Sum(a(n,m,q),{m,0,q}]
%e {1},
%e {1, 2},
%e {1, 3, 4},
%e {1, 4, 6, 7},
%e {1, 5, 7, 9, 11},
%e {1, 6, 9, 12, 14, 16},
%e {1, 7, 10, 13, 16, 19, 21},
%e {1, 8, 12, 15, 18, 21, 24, 27},
%e {1, 9, 13, 17, 21, 24, 27, 30, 33},
%e {1, 10, 15, 20, 25, 28, 32, 35, 39, 42},
%e {1, 11, 16, 21, 26, 30, 34, 38, 42, 46, 50},
%e {1, 12, 18, 23, 28, 33, 38, 42, 46, 50, 54, 58},
%e {1, 13, 19, 25, 31, 37, 42, 47, 51, 55, 59, 63, 67}
%t a[0, 0, q_] := 1; a[1, 0, q_] := 1; a[1, 1, q_] = q;
%t a[n_, 0, q_] := 1; a[n_, n_, q_] := q;
%t a[n_, m_, q_] := a[n, m, q] = Floor[(a[n - 1, m - 1, q] + a[n - 1, m, q])/2]
%t Table[Table[Sum[a[n, m,q], {m, 0, n}], {n, 0, q}], {q, 0, 12}];
%t Flatten[%]
%K nonn,tabl,uned
%O 0,3
%A _Roger L. Bagula_, Mar 30 2010