A226377 Lucas numbers differences triangle T(n,k), k<=n, where column k+1 holds the k-th differences of A000204, read by rows.

1, 3, 2, 4, 1, -1, 7, 3, 2, 3, 11, 4, 1, -1, -4, 18, 7, 3, 2, 3, 7, 29, 11, 4, 1, -1, -4, -11, 47, 18, 7, 3, 2, 3, 7, 18, 76, 29, 11, 4, 1, -1, -4, -11, -29, 123, 47, 18, 7, 3, 2, 3, 7, 18, 47, 199, 76, 29, 11, 4, 1, -1, -4, -1, -29, -76, -199

Offset: 1

Author

Richard R. Forberg, Jul 31 2013

Keywords

• sign
• tabl

Comments

Consecutive columns (i.e. k =1,2,3...) shift the Lucas sequence (A000204) down by 2 indices.

Diagonal (n=k) produces A061084, and Lucas numbers at increasingly negative indices for n=k>2.

Row sums equal A203976(n) for n=>1, which equals Lucas numbers A000204(n) if n is odd, and 5 * A000045(2*n) (Fibonacci) if n is even.

Compare A227431 which is a differences triangle for the Fibonacci sequence A000045.

Examples

			Triangle begins:
1;
3,  2;
4,  1, -1;
7,  3,  2,  3;
11,  4,  1, -1, -4;
18,  7,  3,  2,  3,  7;
29, 11,  4,  1, -1, -4, -11;
47, 18,  7,  3,  2,  3,   7,  18;
76, 29, 11,  4,  1, -1,  -4, -11, -29;
...
		

Crossrefs

Cf. A000204, A203976

Formula

T(n,1) = A000204(n) for n>0, T(n,k) = T(n,k-1) - T(n-1,k-1).