OFFSET
0,5
COMMENTS
The ratio of consecutive row sums seems to converge to a value near 2, but it is not yet certain that the limit exists.
EXAMPLE
Triangle begins:
1;
1,1;
1,2,1;
1,3,1,1;
1,4,1,4,2;
1,5,1,9,4,1;
1,6,1,16,6,1,1;
1,7,1,25,8,1,6,3;
1,8,1,36,10,1,17,8,1;
1,9,1,49,12,1,36,15,2,1;
1,10,1,64,14,1,65,24,3,1,1; ...
which when flattened (read by rows) forms diagonal 0:
1,1,1,1,2,1,1,3,1,1,1,4,1,4,2,1,5,1,9,4,1,...
The matrix square, T^2 = A105537, begins:
1;
2,1;
4,4,1;
6,8,2,1;
12,26,7,12,4; ...
which when flattened (read by rows) forms diagonal 1:
1,2,1,4,4,1,6,8,2,1,12,26,7,12,4,...
The matrix cube, T^3 = A105539, begins:
1;
3,1;
9,6,1;
17,15,3,1;
61,92,23,28,8; ...
which when flattened (read by rows) forms diagonal 2:
1,3,1,9,6,1,17,15,3,1,61,92,23,28,8,...
PROG
(PARI) {T(n, k)=local(A=matrix(n+1, n+1, r, c, if(r>=c, 1)), B, TK=floor((sqrt(8*k+1)-1)/2), TC); B=A; for(r=0, n, for(c=0, r, TC=floor((sqrt(8*c+1)-1)/2); B[r+1, c+1]=(A^(r-c+1))[TC+1, c-TC*(TC+1)/2+1]); A=B); return((A^(n-k+1))[TK+1, k-TK*(TK+1)/2+1])}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 12 2005
STATUS
approved