cp's OEIS Frontend

A123937 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.

%I A123937 #8 Jan 21 2013 09:58:11
%S A123937 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,
%T A123937 12,32,80,32,64,128,1,4,16,32,80,192,64,128,256,1,6,16,56,80,192,448,
%U A123937 128,256,512,1,6,24,56,176,192,448,1024,256,512,1024
%N A123937 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.
%C A123937 Comments from R. J. Mathar, Oct 31 2006:
%C A123937 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:
%C A123937 A122510(d,n)=b(0,d,n) [with an auxiliary, virtual A122510(0,n)=1].
%C A123937 Seeds to construct two sides of b(.,.,.):
%C A123937 b(x,0,n)=A123937(n,x) for x<=n; b(n,y,n)=A123937(n,n) for y>=0.
%C A123937 Recurrence within the b(.,.,.) : b(x,y,n)=b(x,y-1,n)+b(x+1,y-1,n) for x<n.
%C A123937 Graphical support as if the array were built top-down and left-to-right from the seeds:
%C A123937 Triangle stump ("stump" means cut-off/finiteness at the bottom and top)
%C A123937 ...................b(n,0,n)...b(n,1,n)...b(n,2,n)....
%C A123937 ..............................
%C A123937 .............b(2,0,n)...b(2,1,n)....
%C A123937 .........b(1,0,n)...b(1,1,n)....
%C A123937 ...b(0,0,n)..b(0,1,n)...b(0,2,n)....
%C A123937 equals triangle stump (note that the top line is constant) T(x,y)=A123937(x,y)
%C A123937 ...................T(n,n)...T(n,n)...T(n,n)....
%C A123937 ..............................
%C A123937 .............T(n,2).....b(2,1,n)....
%C A123937 .........T(n,1).....b(1,1,n)....
%C A123937 ...T(n,0)....b(0,1,n)...b(0,2,n)....
%C A123937 equals triangle stump
%C A123937 ...................T(n,n)...T(n,n)...T(n,n)....
%C A123937 ..............................
%C A123937 .............T(n,2).....b(2,1,n)....
%C A123937 .........T(n,1).....b(1,1,n)....
%C A123937 ...T(n,0)...A122510(1,n).A122510(2,n).A122510(3,n)....
%e A123937 Triangle begins:
%e A123937 1
%e A123937 1 2
%e A123937 1 2 4
%e A123937 1 2 4 8
%e A123937 1 4 4 8 16
%e A123937 1 4 12 8 16 32
%e A123937 1 4 12 32 16 32 64
%e A123937 1 4 12 32 80 32 64 128
%K A123937 nonn,tabl
%O A123937 0,3
%A A123937 _David W. Wilson_, Oct 30 2006