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A010913
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Pisot sequence E(3,17), a(n) = floor( a(n-1)^2/a(n-2)+1/2 ).
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2
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3, 17, 96, 542, 3060, 17276, 97536, 550664, 3108912, 17552144, 99095040, 559465952, 3158605632, 17832701888, 100679000064, 568408596608, 3209093579520, 18117744283904, 102288278544384, 577494182698496, 3260388539102208, 18407342869216256, 103923280137093120
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OFFSET
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0,1
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LINKS
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FORMULA
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Up to n=10^5, a(n) = 6a(n-1) - 2a(n-2). - Ralf Stephan, Sep 03 2013
Conjecture: If p[i]=fibonacci(2i+2) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
a(n) = (((3-sqrt(7))^n*(-8+3*sqrt(7))+(3+sqrt(7))^n*(8+3*sqrt(7))))/(2*sqrt(7)).
a(n) = 6*a(n-1)-2*a(n-2) for n>1.
G.f.: (3-x) / (1-6*x+2*x^2). (End)
Theorem: a(n) = 6 a(n - 1) - 2 a(n - 2) for n>=2. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - N. J. A. Sloane, Sep 09 2016
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MATHEMATICA
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RecurrenceTable[{a[1] == 3, a[2] == 17, a[n] == Floor[a[n-1]^2/a[n-2]+1/2]}, a, {n, 40}] (* Vincenzo Librandi, Aug 09 2016 *)
LinearRecurrence[{6, -2}, {3, 17}, 30] (* Harvey P. Dale, Nov 12 2021 *)
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PROG
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(PARI) Vec((3-x)/(1-6*x+2*x^2) + O(x^25)) \\ Jinyuan Wang, Mar 10 2020~
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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