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A123937
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Triangle read by rows: T(x, y) = 0 if y > x, = 1 if y = 0, or = 2*Sum_{k >= 1, x-k^2 >= y} T(x-k^2, y-1) otherwise. The zeros are omitted from the sequence.
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2
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1, 1, 2, 1, 2, 4, 1, 2, 4, 8, 1, 4, 4, 8, 16, 1, 4, 12, 8, 16, 32, 1, 4, 12, 32, 16, 32, 64, 1, 4, 12, 32, 80, 32, 64, 128, 1, 4, 16, 32, 80, 192, 64, 128, 256, 1, 6, 16, 56, 80, 192, 448, 128, 256, 512, 1, 6, 24, 56, 176, 192, 448, 1024, 256, 512, 1024
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OFFSET
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0,3
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COMMENTS
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Comments from R. J. Mathar, Oct 31 2006:
This sequence provides the seeds for the construction of columns (vertical recurrence) of A122510 insofar as each row of A123937 provides two sides of auxiliary arrays b(.,.,.) from which a column of A122510 emerges as the third side:
Seeds to construct two sides of b(.,.,.):
Recurrence within the b(.,.,.) : b(x,y,n)=b(x,y-1,n)+b(x+1,y-1,n) for x<n.
Graphical support as if the array were built top-down and left-to-right from the seeds:
Triangle stump ("stump" means cut-off/finiteness at the bottom and top)
...................b(n,0,n)...b(n,1,n)...b(n,2,n)....
..............................
.............b(2,0,n)...b(2,1,n)....
.........b(1,0,n)...b(1,1,n)....
...b(0,0,n)..b(0,1,n)...b(0,2,n)....
equals triangle stump (note that the top line is constant) T(x,y)=A123937(x,y)
...................T(n,n)...T(n,n)...T(n,n)....
..............................
.............T(n,2).....b(2,1,n)....
.........T(n,1).....b(1,1,n)....
...T(n,0)....b(0,1,n)...b(0,2,n)....
equals triangle stump
...................T(n,n)...T(n,n)...T(n,n)....
..............................
.............T(n,2).....b(2,1,n)....
.........T(n,1).....b(1,1,n)....
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LINKS
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EXAMPLE
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Triangle begins:
1
1 2
1 2 4
1 2 4 8
1 4 4 8 16
1 4 12 8 16 32
1 4 12 32 16 32 64
1 4 12 32 80 32 64 128
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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