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A065220
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a(n) = Fibonacci(n) - n.
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13
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0, 0, -1, -1, -1, 0, 2, 6, 13, 25, 45, 78, 132, 220, 363, 595, 971, 1580, 2566, 4162, 6745, 10925, 17689, 28634, 46344, 75000, 121367, 196391, 317783, 514200, 832010, 1346238, 2178277, 3524545, 5702853, 9227430, 14930316, 24157780, 39088131, 63245947, 102334115, 165580100, 267914254
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,7
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COMMENTS
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E(n) = Fib(n+4)-(n+4): cost of maximum height Huffman tree of size n for Fibonacci sequence (Fibonacci sequence is minimizing absolutely ordered sequence of Huffman tree). - Alex Vinokur (alexvn(AT)barak-online.net), Oct 26 2004
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REFERENCES
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Vinokur A.B, Huffman trees and Fibonacci numbers, Kibernetika Issue 6 (1986) 9-12 (in Russian); English translation in Cybernetics 21, Issue 6 (1986), 692-696.
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LINKS
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FORMULA
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a(n) = a(n-1) + a(n-2) + n - 3 = a(n-1) + A000071(n-2).
G.f.: x^2*(2x-1)/((1-x-x^2)*(1-x)^2).
a(n) = Sum_{i=0..n} (i - 2)*F(n-i) for F(n) the Fibonacci sequence A000045. - Greg Dresden, Jun 01 2022
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MAPLE
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a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+a[n-2] od: seq(a[n]-n, n=0..42); # Zerinvary Lajos, Mar 20 2008
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MATHEMATICA
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Table[Fibonacci[n]-n, {n, 0, 50}] (* or *) LinearRecurrence[{3, -2, -1, 1}, {0, 0, -1, -1}, 50] (* Harvey P. Dale, May 29 2017 *)
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PROG
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(PARI) { for (n=0, 300, write("b065220.txt", n, " ", fibonacci(n) - n) ) } \\ Harry J. Smith, Oct 14 2009
(Haskell)
a065220 n = a065220_list !! n
a065220_list = zipWith (-) a000045_list [0..]
(Magma) [Fibonacci(n) - n: n in [0..50]]; // G. C. Greubel, Jul 09 2019
(Sage) [fibonacci(n) - n for n in (0..50)] # G. C. Greubel, Jul 09 2019
(GAP) List([0..50], n-> Fibonacci(n) - n) # G. C. Greubel, Jul 09 2019
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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