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Search: id:A139764
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| A139764 |
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Smallest term in Zeckendorf representation of n. |
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+0
1
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| 1, 2, 3, 1, 5, 1, 2, 8, 1, 2, 3, 1, 13, 1, 2, 3, 1, 5, 1, 2, 21, 1, 2, 3, 1, 5, 1, 2, 8, 1, 2, 3, 1, 34, 1, 2, 3, 1, 5, 1, 2, 8, 1, 2, 3, 1, 13, 1, 2, 3, 1, 5, 1, 2, 55, 1, 2, 3, 1, 5, 1, 2, 8, 1, 2, 3, 1, 13, 1, 2, 3, 1, 5, 1, 2, 21, 1, 2, 3, 1, 5, 1, 2, 8, 1, 2, 3, 1, 89
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Also called a "Fibonacci fractal".
a(n) is also the number of matches to take away to win in a certain match game (see ROCHER et al.).
The frequencies of occurrences of the values in this sequence and A035614 are related by the golden ratio.
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REFERENCES
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A. J. Schwenk, Take-Away Games, The Fibonacci Quarterly, v 8, no 3 (1970), 225-234.
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LINKS
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Steve Witham, Table of n, a(n) for n=1..9999
Author?, Title? (see p. 24)
Alex Bogomolny, Theory of Take-Away Games
Sylvain ROCHER, Elodie PRIVAT, Laurent ORBAN, Alexandre MOTHE and Laurent THOUY, La stratgie des allumettes
Eric Weisstein's World of Math, Wythoff Array
Wikipedia, Zeckendorf's theorem
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FORMULA
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a(n) = n if n is a Fibonacci number, else a( n - (largest Fibonacci number < n) )
a(n) = the value of the (exactly one) digit that turns on between the Fibonacci-base representations of n-1 and n. E.g. from 6 (1001) to 7 (1010), the two's digit turns on.
a(n) = top element of the column of the Wythoff array that contains n
a(n) = Fibonacci( A035614(n) + 1 )
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EXAMPLE
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The Zeckendorf representation of 7 = 5 + 2, so a(7) = 2.
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MAPLE
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(Maple program from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 22 2008)
A000045 := proc(n) combinat[fibonacci](n) ; end:
A087172 := proc(n)
local a, i ;
a := 0 ;
for i from 0 do
if A000045(i) <= n then
a := A000045(i) ;
else
RETURN(a) ;
fi ;
od:
end:
A139764 := proc(n)
local nResid, prevF ;
nResid := n ;
while true do
prevF := A087172(nResid) ;
if prevF = nResid then
RETURN(prevF) ;
else
nResid := nResid-prevF ;
fi ;
od:
end:
seq(A139764(n), n=1..120) ;
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CROSSREFS
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Cf. A035614, A107017, A014417, A006519.
Sequence in context: A125182 A080063 A140706 this_sequence A089026 A080305 A053815
Adjacent sequences: A139761 A139762 A139763 this_sequence A139765 A139766 A139767
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KEYWORD
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nonn,nice
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AUTHOR
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Steve Witham (sw(AT)tiac.net), May 15 2008
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EXTENSIONS
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More terms from T. D. Noe and R. J. Mathar, May 22 2008
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