|
|
A178521
|
|
The cost of all leaves in the Fibonacci tree of order n.
|
|
4
|
|
|
0, 0, 3, 7, 17, 35, 70, 134, 251, 461, 835, 1495, 2652, 4668, 8163, 14195, 24565, 42331, 72674, 124354, 212155, 360985, 612743, 1037807, 1754232, 2959800, 4985475, 8384479, 14080601, 23614931, 39556030, 66181310, 110608187, 184670693, 308030923, 513334855
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node. In a Fibonacci tree the cost of a left (right) edge is defined to be 1 (2). The cost of a leaf of a Fibonacci tree is defined to be the sum of the costs of the edges that form the path from the root to this leaf.
|
|
REFERENCES
|
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = n*F(n+1) - F(n), where F(k) = A000045(k).
a(n) = (2^(-1-n)*(2*sqrt(5)*((1-sqrt(5))^n - (1+sqrt(5))^n) + (-(1-sqrt(5))^n*(-5+sqrt(5)) + (1+sqrt(5))^n*(5+sqrt(5)))*n))/5. - Colin Barker, Jul 26 2017
a(n) = (-n*sin(c*(-n - 1)) - sin(c*n)*i)/((-i)^(-n)*sqrt(5/4)) where c = arccos(i/2). - Peter Luschny, May 16 2022
|
|
MAPLE
|
with(combinat); seq(n*fibonacci(n+1)-fibonacci(n), n = 0 .. 35);
|
|
MATHEMATICA
|
Table[n Fibonacci[n + 1] - Fibonacci[n], {n, 0, 40}] (* Harvey P. Dale, Apr 21 2011 *)
Table[(n - 1) Fibonacci[n] + n Fibonacci[n - 1], {n, 0, 40}] (* Bruno Berselli, Dec 06 2013 *)
|
|
PROG
|
(PARI) concat(vector(2), Vec(x^2*(x+3)/(x^2+x-1)^2 + O(x^50))) \\ Colin Barker, Jul 26 2017
(Julia) # The function 'fibrec' is defined in A354044.
n < 2 && return BigInt(0)
a, b = fibrec(n - 1)
a*n + (n - 1)*b
end
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|