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%I #41 Sep 08 2022 08:44:47
%S 2,5,11,21,38,66,112,187,309,507,828,1348,2190,3553,5759,9329,15106,
%T 24454,39580,64055,103657,167735,271416,439176,710618,1149821,1860467,
%U 3010317,4870814,7881162,12752008,20633203,33385245,54018483,87403764,141422284
%N Convolution of natural numbers >= 2 and Fibonacci numbers.
%C Minimal cost of maximum height Huffman tree of size n for strictly "worst case height" sequences. (A strictly "worst case height" sequence generates only maximum height Huffman trees; a non-strictly "worst case height" sequence can generate also non-maximum height Huffman trees.) - Alex Vinokur (alexvn(AT)barak-online.net), Oct 26 2004
%C Record-positions for A107910: A107910(a(n+2)) = A005578(n), A107910(m) < A005578(n) for m < a(n+2). - _Reinhard Zumkeller_, May 28 2005
%H Colin Barker, <a href="/A023548/b023548.txt">Table of n, a(n) for n = 1..1000</a>
%H N.-N. Cao, F.-Z. Zhao, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Cao2/cao5r.html">Some Properties of Hyperfibonacci and Hyperlucas Numbers</a>, J. Int. Seq. 13 (2010) # 10.8.8
%H Ligia L. Cristea, Ivica Martinjak, and Igor Urbiha, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Urbiha/urbiha4.html">Hyperfibonacci Sequences and Polytopic Numbers</a>, Journal of Integer Sequences, Volume 19, 2016, Issue 7, #16.7.6.
%H A. B. Vinokur, <a href="http://dx.doi.org/10.1007/BF01068684">Huffman trees and Fibonacci numbers</a>, Kibernetika Issue 6 (1986) 9-12 (in Russian); English translation in Cybernetics 21, Issue 6 (1986), 692-696.
%H Alex Vinokur, <a href="http://arXiv.org/abs/cs/0410013">Fibonacci connection between Huffman codes and Wythoff array</a>, arXiv:cs/0410013 [cs.DM], 2004-2005.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F From _Wolfdieter Lang_: (Start)
%F Convolution of natural numbers n >= 1 with Lucas numbers (A000032).
%F a(n) = 4*(F(n+1) - 1) + 3*F(n) - n, F(n)=A000045 (Fibonacci).
%F G.f.: x*(2-x)/((1-x-x^2)*(1-x)^2). (End)
%F For n >= 1, a(n) = L(n+3) - (n+4), where L(n) are Lucas numbers. - Mario Catalani (mario.catalani(AT)unito.it), Jul 22 2004
%F a(n) = Fib(n+3) + F(n+1) - (n+3) for n > 1. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 26 2004
%F a(n) = (-4 + (2^(-n)*((1-sqrt(5))^n*(-5+2*sqrt(5)) + (1+sqrt(5))^n*(5+2*sqrt(5)))) / sqrt(5) - n). - _Colin Barker_, Mar 11 2017
%F a(n) = Sum_{i=1..n} C(n-i+2,i+1) + C(n-i+1,i). - _Wesley Ivan Hurt_, Sep 13 2017
%t Table[4(Fibonacci[n+1] -1) +3Fibonacci[n] -n, {n, 40}] (* _Vincenzo Librandi_, Sep 16 2017 *)
%o (PARI) a(n) = 4*fibonacci(n+1) + 3*fibonacci(n) - n - 4; \\ _Michel Marcus_, Sep 08 2016
%o (PARI) Vec(x*(2-x) / ((1-x-x^2)*(1-x)^2) + O(x^40)) \\ _Colin Barker_, Mar 11 2017
%o (Magma) [4*(Fibonacci(n+1)-1)+3*Fibonacci(n)-n: n in [1..40]]; // _Vincenzo Librandi_, Sep 16 2017
%o (Sage) [lucas_number2(n+3,1,-1) -n-4 for n in (1..40)] # _G. C. Greubel_, Jul 08 2019
%o (GAP) List([1..40], n-> Lucas(1,-1,n+3)[2] -n-4) # _G. C. Greubel_, Jul 08 2019
%Y Cf. A000032, A000045, A006327.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_
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