A193967 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{L(k+1)*x^(n-k) : 0<=k<=n}, where F=A000032 (Lucas numbers), and q(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

1, 1, 1, 3, 4, 7, 4, 7, 12, 19, 7, 11, 21, 33, 54, 11, 18, 33, 54, 88, 142, 18, 29, 54, 87, 144, 232, 376, 29, 47, 87, 141, 232, 376, 609, 985, 47, 76, 141, 228, 376, 608, 987, 1596, 2583, 76, 123, 228, 369, 608, 984, 1596, 2583, 4180, 6763, 123, 199, 369

Offset

0,4

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Links

Table of n, a(n) for n=0..57.

Example

First six rows:
1
1...1
3...4...7
4...7...12..19
7...11..21..33..54
11..18..33..54..88..142

Mathematica

z = 12;
p[n_, x_] := Sum[LucasL[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193967 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193968 *)

Crossrefs

Cf. A193722, A193968.

Sequence in context: A200681 A161775 A282535 * A109823 A337124 A267447

Adjacent sequences: A193964 A193965 A193966 * A193968 A193969 A193970

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 10 2011

Status

approved