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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
0

%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