0,1

Leftmost column = alternate Lucas numbers: (2, 3, 7, 18...), given the Lucas sequence starting (2, 1, 3, 4, 7, 11, l8,...). Refer to A084534 for a variation of the Lucas polynomials.

Jay Kappraff, "Beyond Measure, A Guided tour Through Nature, Myth and Number", World Scientific, 2002, p. 485 (Table 22.6b).

Diagonals are sequences as f(x), x=1,2,3; Lucas polynomials in the format: (2); (x + 2); (x^2 + 4x + 2); (x^3 + 6x^2 + 9x + 2); (x^4 + 8x^3 + 20x^2 + 16x + 2); (x^5 + 10x^4 + 35x^3 + 50x^2 + 25x + 2);... Diagonals of the triangle are binomial transforms of A118008 rows.

First few rows of the triangle are:

2;

3, 2;

7, 4, 2;

18, 14, 5, 2;

47, 52, 23, 6, 2;

123, 194, 110, 34, 7, 2;

...

For example, 4-th diagonal from the right (18, 52, 110...) = f(x), x=1,2,3...: x^3 + 6x^2 + 9x + 2.

(18, 52, 110...) = binomial transform of 4-th row of A118008: (18, 34, 24, 6).

Cf. A084534, A118008.

Sequence in context: A104565 A144456 A051886 this_sequence A158747 A122697 A129022

Adjacent sequences: A118004 A118005 A118006 this_sequence A118008 A118009 A118010

nonn

Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 09 2006