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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified July 12 04:30 EDT 2020. Contains 335658 sequences. (Running on oeis4.)