0,5

The ratio of consecutive row sums seems to converge to a value near 2, but it is not yet certain that the limit exists.

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,...

(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])}

Cf. A105536 (row sums), A105537 (matrix square), A105539 (matrix cube).

Sequence in context: A165430 A164823 A167269 this_sequence A108888 A124021 A109626

Adjacent sequences: A105532 A105533 A105534 this_sequence A105536 A105537 A105538

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 12 2005