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A065220 a(n) = Fibonacci(n) - n. 12
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)
OFFSET

0,7

COMMENTS

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

REFERENCES

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.

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..300

Albert Frank, International Contest Of Logical Sequences, 2002 - 2003. Item 7

Albert Frank, Solutions of International Contest Of Logical Sequences, 2002 - 2003.

A. B. Vinokur, Huffman trees and Fibonacci numbers, Cybernetics 21, Issue 6 (1986), 692-696; also at Research Gate.

Alex Vinokur, Fibonacci connection between Huffman codes and Wythoff array, arXiv:cs/0410013 [cs.DM], 2004-2005.

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

FORMULA

a(n) = A000045(n) - A001477(n) = A000126(n-3) - 2 = A001924(n-4) - 1.

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).

MAPLE

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

MATHEMATICA

lst={}; Do[f=Fibonacci[n]-n; AppendTo[lst, f], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Mar 21 2009 *)

Table[Fibonacci[n]-n, {n, 0, 50}] (* or *) LinearRecurrence[{3, -2, -1, 1}, {0, 0, -1, -1}, 50] (* Harvey P. Dale, May 29 2017 *)

PROG

(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..]

-- Reinhard Zumkeller, Nov 06 2012

(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

CROSSREFS

Cf. A002062, A045925.

Sequence in context: A000135 A281865 A267698 * A048094 A181522 A031872

Adjacent sequences:  A065217 A065218 A065219 * A065221 A065222 A065223

KEYWORD

easy,sign

AUTHOR

Henry Bottomley, Oct 22 2001

STATUS

approved

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Last modified October 18 22:16 EDT 2019. Contains 328211 sequences. (Running on oeis4.)