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%I #36 Mar 05 2024 08:00:27
%S 1,2,3,5,9,15,24,39,64,104,168,272,441,714,1155,1869,3025,4895,7920,
%T 12815,20736,33552,54288,87840,142129,229970,372099,602069,974169,
%U 1576239,2550408,4126647,6677056,10803704,17480760,28284464,45765225
%N Positive 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.
%C 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_, Dec 19 2007
%C If the formula is correct, the bisections give A059840 and A064831. - _David Wasserman_, Dec 19 2007
%H G. C. Greubel, <a href="/A097083/b097083.txt">Table of n, a(n) for n = 1..1000</a>
%H Andreas M. Hinz and Paul K. Stockmeyer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Hinz/hinz5.html">Precious Metal Sequences and Sierpinski-Type Graphs</a>, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
%F 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.
%F 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). - _Gary Detlefs_, Mar 12 2011
%F From _R. J. Mathar_, Mar 13 2011: (Start)
%F If the partial sum assumption is correct: a(n)= +2*a(n-1) -a(n-2) +a(n-3) -a(n-5).
%F G.f.: x/( (x-1)*(x^2+1)*(x^2+x-1) ).
%F a(n) = A000032(n+3)/5 -(-1)^n*A112030(n)/10 - 1/2. (End)
%F Conjecture: a(n) = floor(F(n+3)/sqrt(5)), where F(n) = A000045(n) are Fibonacci numbers. - _Vladimir Reshetnikov_, Nov 05 2015
%t a=b=c=d=0;Table[e=a+b+d+1;a=b;b=c;c=d;d=e,{n,100}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 26 2011 *)
%t CoefficientList[Series[x/((x - 1)*(x^2 + 1)*(x^2 + x - 1)), {x,0,50}], x] (* _G. C. Greubel_, Mar 05 2017 *)
%o (PARI) x='x+O('x^50); Vec(x/((x - 1)*(x^2 + 1)*(x^2 + x - 1))) \\ _G. C. Greubel_, Mar 05 2017
%Y Cf. A073364, A000045.
%K nonn
%O 1,2
%A _John W. Layman_, Jul 23 2004
%E a(9) from _Ray Chandler_, Jul 29 2004
%E More terms from _David Wasserman_, Dec 19 2007
%E Terms > 90000 assuming the partial sums formula by _Vladimir Joseph Stephan Orlovsky_, Feb 26 2011
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