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%I #16 Sep 08 2022 08:45:54
%S 0,0,3,8,22,49,104,208,403,760,1406,2561,4608,8208,14499,25432,44342,
%T 76913,132808,228416,391475,668840,1139518,1936513,3283392,5555424,
%U 9381699,15815528,26618518,44733745,75073256,125827696,210642643
%N The sum of the costs of all nodes in the Fibonacci tree of order n.
%C 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 node in 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.
%C A178525 is the 1-sequence of reduction of the odd number sequence (2n-1) by x^2 -> x+1; as such it is related to 0-sequence of this reduction, A192304. See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]". - _Clark Kimberling_, Jun 27 2011
%D D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
%H G. C. Greubel, <a href="/A178525/b178525.txt">Table of n, a(n) for n = 0..1000</a>
%H Y. Horibe, <a href="http://www.fq.math.ca/Scanned/20-2/horibe.pdf">An entropy view of Fibonacci trees</a>, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,1,1).
%F a(n) = 3 + (2*n-3)*F(n-1) + (2*n-5)*F(n), where F(k)=A000045(k) are the Fibonacci numbers.
%F a(n) = a(n-1) + a(n-2) + 2*F(n+1) + 2*F(n-1) - 3 (n>=2), F(0)=0, F(1)=0.
%F G.f.: z^2*(3-z+z^2)/((1-z)*(1-z-z^2)^2).
%p with(combinat): seq(3+(2*n-3)*fibonacci(n-1)+(2*n-5)*fibonacci(n), n = 0 .. 32);
%t Table[3 +(2*n-3)*Fibonacci[n-1] +(2*n-5)*Fibonacci[n], {n,0,40}] (* _G. C. Greubel_, Jan 30 2019 *)
%o (PARI) a(n) = 3+(2*n-3)*fibonacci(n-1) + (2*n-5)*fibonacci(n); \\ _Michel Marcus_, Jan 21 2019
%o (Magma) [3 +(2*n-3)*Fibonacci(n-1) +(2*n-5)*Fibonacci(n): n in [0..40]]; // _G. C. Greubel_, Jan 30 2019
%o (Sage) [3 +(2*n-3)*fibonacci(n-1) +(2*n-5)*fibonacci(n) for n in range(40)] # _G. C. Greubel_, Jan 30 2019
%o (GAP) List([0..40], n -> 3 +(2*n-3)*Fibonacci(n-1) +(2*n-5)*Fibonacci(n)); # _G. C. Greubel_, Jan 30 2019
%Y Cf. A000045, A094584, A178521.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Jun 15 2010
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