A105540 Triangular matrix T, read by rows, where column n equals T^(n+1) when flattened (as read by rows), for n>=0.

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 4, 1, 4, 1, 1, 4, 9, 1, 5, 1, 1, 1, 6, 16, 1, 6, 1, 1, 6, 1, 8, 25, 1, 7, 1, 3, 8, 21, 1, 10, 36, 1, 8, 1, 1, 6, 21, 52, 1, 12, 49, 1, 9, 1, 2, 1, 9, 40, 105, 1, 14, 64, 1, 10, 1, 4, 13, 1, 12, 65, 186, 1, 16, 81, 1, 11, 1, 1, 14, 51, 1, 15, 96, 301, 1, 18, 100, 1

Offset

0,5

Comments

The ratio of consecutive row sums seems to converge to a value near sqrt(2).

Links

Table of n, a(n) for n=0..88.

Example

Triangle T begins:
1;
1,1;
1,2,1;
1,1,3,1;
2,4,1,4,1;
1,4,9,1,5,1;
1,1,6,16,1,6,1;
1,6,1,8,25,1,7,1;
3,8,21,1,10,36,1,8,1;
1,6,21,52,1,12,49,1,9,1;
2,1,9,40,105,1,14,64,1,10,1;
4,13,1,12,65,186,1,16,81,1,11,1;
1,14,51,1,15,96,301,1,18,100,1,12,1; ...
which when flattened (read by rows) forms column 0:
1,1,1,1,2,1,1,1,3,1,2,4,1,4,1,1,4,9,1,5,1,...
The matrix square, T^2 = A105542, begins:
1;
2,1;
4,4,1;
6,8,6,1;
13,14,14,8,1;
26,47,26,22,10,1; ...
which when flattened (read by rows) forms column 1:
1,2,1,4,4,1,6,8,6,1,13,14,14,8,1,...
The matrix cube, T^3 = A105545, begins:
1;
3,1;
9,6,1;
21,21,9,1;
51,54,39,12,1;
142,165,111,63,15,1; ...
which when flattened (read by rows) forms column 2:
1,3,1,9,6,1,21,21,9,1,51,54,39,12,1,...

Prog

(PARI) {T(n, k)=local(A=matrix(n+1, n+1, r, c, if(r>=c, 1)), B, TK=floor((sqrt(8*(n-k)+1)-1)/2), TC); B=A; for(r=0, n, for(c=0, r, TC=floor((sqrt(8*(r-c)+1)-1)/2); B[r+1, c+1]=(A^(c+1))[TC+1, r-c-TC*(TC+1)/2+1]); A=B); return((A^(k+1))[TK+1, n-k-TK*(TK+1)/2+1])}

Crossrefs

Cf. A105541 (row sums), A105542 (matrix square), A105545 (matrix cube).

Sequence in context: A204178 A365466 A095136 * A356958 A057043 A325307

Adjacent sequences: A105537 A105538 A105539 * A105541 A105542 A105543

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 13 2005

Status

approved