A113506 Maximum element in the continued fraction expansion of F(n+1)^5/F(n)^5 where F=A000045.
1, 32, 7, 12, 17, 46, 3042, 319, 835, 2188, 5730, 375131, 39282, 102845, 269253, 704915, 46137317, 4831563, 12649196, 33116027, 86698886, 5674515856, 594243013, 1555748409, 4073002214, 10663258234, 697919312217, 73087059232
Offset: 1
Keywords
Crossrefs
Cf. A113500.
Programs
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Mathematica
Max[ContinuedFraction[#[[2]]/#[[1]]]]&/@Partition[Fibonacci[ Range[ 30]]^5,2,1] (* Harvey P. Dale, May 27 2018 *)
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PARI
a(n)=vecmax(contfrac(fibonacci(n+1)^5/fibonacci(n)^5))
Formula
5*a(5n)=F(10n+1)-(-1)^n-5; 5*a(5n+1)=F(10n+3)-2*(-1)^n-5; a(5n+2)=5*F(10n+5)+7*(-1)^n-1; 5a(5n+3)=F(10n+7)-3*(-1)^n-5; 5a(5n+4)=F(10n+9)+(-1)^n-5.
Empirical g.f.: x*(6*x^19 -5*x^17 +x^16 -x^15 -732*x^14 -x^13 +593*x^12 -107*x^11 +118*x^10 -743*x^9 +94*x^8 -327*x^7 +786*x^6 +93*x^5 -5*x^4 -5*x^3 +25*x^2 -31*x -1) / ((x -1) * (x +1) * (x^2 -3*x +1) * (x^4 -x^3 +x^2 -x +1) * (x^4 -x^3 +6*x^2 +4*x +1) * (x^4 +4*x^3 +6*x^2 -x +1)). - Colin Barker, Jun 17 2013