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.

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

Offset

0,3

Comments

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:

A122510(d,n)=b(0,d,n) [with an auxiliary, virtual A122510(0,n)=1].

Seeds to construct two sides of b(.,.,.):

b(x,0,n)=A123937(n,x) for x<=n; b(n,y,n)=A123937(n,n) for y>=0.

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)....

...T(n,0)...A122510(1,n).A122510(2,n).A122510(3,n)....

Links

Table of n, a(n) for n=0..65.

Example

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

Crossrefs

Sequence in context: A059268 A300653 A256009 * A138882 A074634 A152036

Adjacent sequences: A123934 A123935 A123936 * A123938 A123939 A123940

Keyword

nonn,tabl

Author

David W. Wilson, Oct 30 2006

Status

approved