A118007 Triangle, diagonals generated from Lucas polynomials.

2, 3, 2, 7, 4, 2, 18, 14, 5, 2, 47, 52, 23, 6, 2, 123, 194, 110, 34, 7, 2, 322, 724, 527, 198, 47, 8, 2, 843, 2702, 2525, 1154, 322, 62, 9, 2, 2207, 10084, 12098, 6726, 2207, 488, 79, 10, 2, 5778, 37634, 57965, 39202, 15127, 3842, 702, 98, 11, 2

Offset

0,1

Comments

Leftmost column = A005248, bisection of Lucas sequence A000032.

Refer to A084534 for a variation of the Lucas polynomials.

References

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

Links

Table of n, a(n) for n=0..54.

Formula

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.

Example

First few rows of the triangle:
    2;
    3,   2;
    7,   4,   2;
   18,  14,   5,  2;
   47,  52,  23,  6, 2;
  123, 194, 110, 34, 7, 2;
  ...
For example, 4th 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 4th row of A118008: (18, 34, 24, 6).

Prog

(PARI) TLucas(n, k) = binomial(n-k, k) + binomial(n-k-1, k-1) + (n==0); \\ A084534
pol(n) = Pol(vector(n+1, k, TLucas(2*n, k-1)));
T(n, k) = subst(pol(n-k), x, k+1);
trgT(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Aug 12 2022

Crossrefs

Cf. A000032, A005248, A034807, A118008.

Cf. A084534.

Sequence in context: A379228 A319374 A205687 * A195401 A158747 A260724

Adjacent sequences: A118004 A118005 A118006 * A118008 A118009 A118010

Keyword

nonn,tabl

Author

Gary W. Adamson, Apr 09 2006

Extensions

More terms from Michel Marcus, Aug 12 2022

Status

approved