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COMMENTS
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Define a map Q from a sequence c(1),c(2),c(3),... to a sequence d(1),d(2),d(3),... as follows:
Let A(0,k) be the starting sequence c(1),c(2), ... For m >= 1, define
A(m,k) = A(m-1,k) + A(m-1,k-A(m-1,m)) for k > A(m-1,m);
A(m,k) = A(m-1,k) for k <= A(m-1,m).
For example:
A(0,k):_1,1,1,1,1,1,1,1,1,1,...
+_______0,1,1,1,1,1,1,1,1,1,...
=A(1,k):1,2,2,2,2,2,2,2,2,2,...
+_______0,0,1,2,2,2,2,2,2,2,...
=A(2,k):1,2,3,4,4,4,4,4,4,4,...
+_______0,0,0,1,2,3,4,4,4,4,...
=A(3,k):1,2,3,5,6,7,8,8,8,8,...
+_______0,0,0,0,0,1,2,3,5,6,...
=A(4,k):1,2,3,5,6,8,10,11,13,14,...
+_______0,0,0,0,0,0,1,_2,_3,_5,...
=A(5,k):1,2,3,5,6,8,11,13,16,19,...
+_______0,0,0,0,0,0,0,_0,_1,_2,...
=A(6,k):1,2,3,5,6,8,11,13,17,21,...
(The numbers of leading 0's in the sequence following each + forms the limit-sequence.)
The limit sequence d(1),d(2),d(3),... shares its first 10 terms with T(6,k), so the limit sequence {T(m,k)} as m -> oo) begins 1,2,3,5,6,8,11,13,17,21,...
If we take the sequence whose g.f. is:
(1+x)(1+x^2)(1+x^3)(1+x^5)(1+x^6)(1+x^8)(1+x^11)...(1+x^d(n))...
we get:
1 1 1 2 1 2 3 2 4 4 3 6 4 5 8 5 8 9 7...
Taking the partial sums we get:
1 2 3 5 6 8 11 13 17 21 24 30 34 39 47 52 60 69 76...
the original sequence. More generally, Q appears to take sequence c to a sequence d such that if we take the Weigh transform of the characteristic function of d and convolve it with sequence c, we get the sequence d.
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