|
|
A193969
|
|
Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers) and q(n,x)=sum{L(k+1)*x^(n-k) : 0<=k<=n}, where F=A000032 (Lucas numbers).
|
|
2
|
|
|
1, 1, 3, 1, 4, 7, 2, 7, 12, 21, 3, 11, 19, 33, 54, 5, 18, 31, 54, 88, 144, 8, 29, 50, 87, 142, 232, 376, 13, 47, 81, 141, 230, 376, 609, 987, 21, 76, 131, 228, 372, 608, 985, 1596, 2583, 34, 123, 212, 369, 602, 984, 1594, 2583, 4180, 6765, 55, 199, 343, 597
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
|
|
LINKS
|
|
|
EXAMPLE
|
First six rows:
1
1...3
1...4...7
2...7...12...21
3...11..19...33...54
5...18..31...54...88...144
|
|
MATHEMATICA
|
z = 12;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := Sum[LucasL[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}]] (* A193969 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193970 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|