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A097083
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Values of k such that there is exactly one permutation p of (1,2,3,...,k) such that i+p(i) is a Fibonacci number for 1<=i<=k.
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5
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1, 2, 3, 5, 9, 15, 24, 39, 64, 104, 168, 272, 441, 714, 1155, 1869, 3025, 4895, 7920, 12815, 20736, 33552, 54288, 87840, 142129, 229970, 372099, 602069, 974169, 1576239, 2550408, 4126647, 6677056, 10803704, 17480760, 28284464, 45765225
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OFFSET
| 1,2
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COMMENTS
| Numbers k such that A097082(k) = 1. If f is a fibonacci number and k < f <= 2k, then a permutation for f-k-1 may be extended to a permutation for k, with p(i) = f-i for f-k < i <= k. This explains the sparseness of this sequence. - David Wasserman (dwasserm(AT)earthlink.net), Dec 19 2007
If the formula is correct, the bisections give A059840 and A064831. - David Wasserman (dwasserm(AT)earthlink.net), Dec 19 2007
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FORMULA
| It appears that {a(n)} satisfies a(1)=1, a(2)=2 and, for n>2, a(n)=F(n+2)-a(n-2)-1, where {F(k)} is the sequence of Fibonacci numbers, i.e, that the sequence is the partial sums of A006498.
If the partial sum assumption is correct: a(n)=floor(phi^(n+3)/5), where phi=(1+sqrt(5))/2 = A001622, and a(n)=a(n-1)+a(n-2)+( (n*(n+1)/2) mod 2). [From Gary Detlefs, Mar 12 2011]
If the partial sum assumption is correct: a(n)= +2*a(n-1) -a(n-2) +a(n-3) -a(n-5), g.f. = x / ( (x-1)*(x^2+1)*(x^2+x-1) ), and a(n) = A000032(n+3)/5 -(-1)^n*A112030(n)/10 -1/2. - R. J. Mathar, Mar 13 2011
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MATHEMATICA
| a=b=c=d=0; Table[e=a+b+d+1; a=b; b=c; c=d; d=e, {n, 100}] (*From Vladimir Joseph Stephan Orlovsky, Feb 26 2011*)
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CROSSREFS
| Cf. A073364.
Sequence in context: A074693 A147322 A143282 * A200047 A147877 A003476
Adjacent sequences: A097080 A097081 A097082 * A097084 A097085 A097086
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KEYWORD
| nonn
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AUTHOR
| John W. Layman (layman(AT)math.vt.edu), Jul 23 2004
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EXTENSIONS
| a(9) from Ray Chandler (rayjchandler(AT)sbcglobal.net), Jul 29 2004
More terms from David Wasserman (dwasserm(AT)earthlink.net), Dec 19 2007
Terms > 90000 assuming the partial sums formula by Vladimir Joseph Stephan Orlovsky, Feb 26 2011
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