login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A166106 a(n) = a(n-1) + a(n-2) + F(n), with a(0) = 0, a(1) = 1, a(2) = a(1) + a(0), a(3) = a(2) + a(1), a(4) = a(3) + a(2) + 2. 4
0, 1, 1, 2, 5, 12, 25, 50, 96, 180, 331, 600, 1075, 1908, 3360, 5878, 10225, 17700, 30509, 52390, 89664, 153000, 260375, 442032, 748775, 1265832, 2136000, 3598250, 6052061, 10164540, 17048641, 28559450, 47786400, 79870428, 133359715, 222457608, 370747675 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Consider the recursive call tree for Fibonacci numbers shown in the Wilson, Abelson et al., and Bloch links. This type of tree is a variant of Fibonacci trees shown/defined elsewhere. Here, let us refer it as a recursive Fibonacci tree. A Fibonacci number F(n) is the weight of the root of a weighted recursive Fibonacci tree of order n in which the leafs have a weight of 1, and the weight of a node equals the sum of the weights of its two children. If instead we weight each leaf by the number of nodes above it (considering the root as the topmost node), then for n > 2, a(n) is the weight of the root of such a weighted tree of order n. For example, a(5) = 2+2+2+3+3.
LINKS
Harold Abelson and Gerald Jay Sussman with Julie Sussman, Tree Recursion from "Structure and Interpretation of Computer Programs", MIT Press, 1996, LaTeX2HTML translation by Ryan Bender.
Bill Wilson, The Prolog Dictionary - memoisation (shows Recursive call tree for Fibonacci number f_6).
FORMULA
For n > 1, a(n) = A067331(n-2).
From Colin Barker, May 25 2014: (Start)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) for n > 6.
G.f.: x*(x^5 + 3*x^4 + 2*x^3 - x^2 - x + 1) / (x^2+x-1)^2. (End)
a(n) = (1/25)*2^(-n-1)*(5*((1 - sqrt(5))^(n+1) + (1 + sqrt(5))^(n+1))*n - (25 + sqrt(5))*(1 + sqrt(5))^n + (sqrt(5) - 25)*(1 - sqrt(5))^n), n > 2. - Ilya Gutkovskiy, Apr 26 2016
MATHEMATICA
a[n_] := a[n] = a[n-1] + a[n-2] + Fibonacci[n]; a[0] = 0; a[1] = 1; a[2] = 1; a[3] = 2; a[4] = 5; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Oct 03 2011 *)
PROG
(PARI) s = 33; a = concat([0, 1, 1, 2, 5], vector(s-5)); for(n=6, s, a[n]=a[n-1]+a[n-2]+fibonacci(n))); for(n=1, s, print1(a[n], ", "))
(PARI) concat(0, Vec(x*(x^5+3*x^4+2*x^3-x^2-x+1)/(x^2+x-1)^2 + O(x^100))) \\ Colin Barker, May 25 2014
CROSSREFS
Cf. A000045.
Sequence in context: A368638 A116730 A240847 * A067331 A116734 A101836
KEYWORD
nonn,easy
AUTHOR
Gerald McGarvey, Oct 06 2009
EXTENSIONS
More terms from Colin Barker, May 25 2014
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 19 08:06 EDT 2024. Contains 371782 sequences. (Running on oeis4.)