

A105535


Triangular matrix T, read by rows, where diagonal n equals T^(n+1) when flattened (as read by rows), for n>=0.


4



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, 1, 11, 1, 81, 16, 1, 106, 35, 4, 1, 12, 4, 1, 12, 1, 100, 18, 1, 161, 48, 5, 1, 61, 26
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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.


LINKS



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^(rc+1))[TC+1, cTC*(TC+1)/2+1]); A=B); return((A^(nk+1))[TK+1, kTK*(TK+1)/2+1])}


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



