

A094584


Dot product of (1,2,...,n) and first n distinct Fibonacci numbers.


6



1, 5, 14, 34, 74, 152, 299, 571, 1066, 1956, 3540, 6336, 11237, 19777, 34582, 60134, 104062, 179320, 307855, 526775, 898706, 1529160, 2595624, 4396224, 7431049, 12537917, 21118814, 35517226, 59646386, 100034456, 167562035, 280348531, 468543802, 782277612
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OFFSET

1,2


COMMENTS

a(n) is the cost of all nonleaf nodes in the Fibonacci tree of order n+2. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n1 and whose right subtree is the Fibonacci tree of order n2; 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 node 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 node.  Emeric Deutsch, Jun 14 2010


REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 14.
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, AddisonWesley, Reading, MA, 1998, p. 417. [From Emeric Deutsch, Jun 14 2010]


LINKS



FORMULA

a(n) = F(2) + 2*F(3) + 3*F(4) + ... + n*F(n+1) = (n+1)*F(n+3)  F(n+5) + 3.
G.f.: x*(1+2*x)/((1x)*(1xx^2)^2).  Colin Barker, Nov 11 2012
a(n) = 3*a(n1)  a(n2)  3*a(n3) + a(n4) + a(n5).
a(n) = Sum_{i=1..n+2} (ni+1) * F(ni+2).
a(n) = (30*(1sqrt(5))^n + (15+7*sqrt(5))*2^n  (15+7*sqrt(5))*(3sqrt(5))^n + 2n*((52*sqrt(5))*2^n + (5+2*sqrt(5))*(3sqrt(5))^n)) / (10*(1sqrt(5))^n). (End)


EXAMPLE

a(4) = (1,2,3,4)*(1,2,3,5) = 1+4+9+20 = 34.


MAPLE



MATHEMATICA

Table[Range[n].Fibonacci[Range[2, n+1]], {n, 40}] (* Harvey P. Dale, Aug 21 2011 *)


PROG

(Magma) I:=[1, 5, 14, 34, 74]; [n le 5 select I[n] else 3*Self(n1)Self(n2)3*Self(n3)+Self(n4)+Self(n5): n in [1..40]]; // Vincenzo Librandi, Mar 11 2015
(Magma) [n*Fibonacci(n+3)Fibonacci(n+4)+3: n in [1..40]]; // G. C. Greubel, Apr 28 2019
(GAP) List([1..40], n>(n+1)*Fibonacci(n+3)Fibonacci(n+5)+3); # Muniru A Asiru, Apr 27 2019
(PARI) {a(n) = n*fibonacci(n+3)  fibonacci(n+4) +3}; \\ G. C. Greubel, Apr 28 2019
(Sage) [n*fibonacci(n+3)  fibonacci(n+4) +3 for n in (1..40)] # G. C. Greubel, Apr 28 2019


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



