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.

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

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

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.

Laurent Bloch, Analyse de l'algorithme de Fibonacci

Bill Wilson, The Prolog Dictionary - memoisation (shows Recursive call tree for Fibonacci number f_6).

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

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

Adjacent sequences: A166103 A166104 A166105 * A166107 A166108 A166109

Keyword

nonn,easy

Author

Gerald McGarvey, Oct 06 2009

Extensions

More terms from Colin Barker, May 25 2014

Status

approved