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 A010914 Pisot sequence E(5,17), a(n) = floor(a(n-1)^2 / a(n-2) + 1/2). 3
 5, 17, 58, 198, 676, 2308, 7880, 26904, 91856, 313616, 1070752, 3655776, 12481600, 42614848, 145496192, 496755072, 1696027904, 5790601472, 19770350080, 67500197376, 230460089344, 786839962624, 2686439671808, 9172078761984, 31315435704320, 106917585293312 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 REFERENCES Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305. D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993. S. B. Ekhad, N. J. A. Sloane, D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT] (2016) Index entries for linear recurrences with constant coefficients, signature (4,-2). FORMULA From Max Alekseyev, Sep 03 2013: (Start) It is not hard to show that this sequence satisfies the following simple linear recurrence relation: a(n) = 4*a(n-1) - 2*a(n-2). Proof: Let a(n) = A010914(n) be the sequence defined by the recurrence a(n) = floor( a(n-1)^2/a(n-2) + 1/2 ) and initial terms a(0)=5 and a(1)=17. Define a sequence b(n) by the recurrence b(n) = 4*b(n-1) - 2*b(n-2) and the same initial terms b(0)=a(0)=5 and b(1)=a(1)=17. We want to prove that a(n)=b(n) for all n. The main trick is that instead of proving that the sequence a() satisfies the recurrence for the sequence b(), we will do it the other way around: prove that b() satisfies the recurrence for a(). This will imply that these two sequences coincide. To prove that b() satisfies the recurrence for a(), it is enough to show that abs( b(n-1)^2 - b(n)*b(n-2) ) < b(n-2)/2. To do this, we will use the explicit formula for b(n) that b(n) = (10+7*sqrt(2))/4 * (2+sqrt(2))^n + (10-7*sqrt(2))/4 * (2-sqrt(2))^n. Computing abs( b(n-1)^2 - b(n)*b(n-2) ), one finds that it equals 2^n times a constant. At the same time, b(n) grows as (2+sqrt(2))^n (up to a constant) which allows one to easily prove (e.g., by induction on n) that abs( b(n-1)^2 - b(n)*b(n-2) ) < b(n-2)/2. (End) For positive n, a(n) equals [1,1;1,3]^n.[1,2].[1.2], which means calculate the n-th power of the specific 2 X 2 matrix, multiply from the right by the column vector [1,2], and finally take the dot product of this column vector with the vector [1,2]. - John M. Campbell, Jul 09 2011 a(n) is the 3rd subdiagonal of array A228405. - Richard R. Forberg, Sep 02 2013 a(n) = A079496(n+3) * A016116(n). - Michael Somos, Sep 03 2013 a(n) - a(n-1)^2 / a(n-2) = 2^floor(n/2) / A143609(n+1) < 1/2 if n>1. - Michael Somos, Sep 03 2013 G.f.: (5-3*x) / (1-4*x+2*x^2). - Colin Barker, Dec 06 2015 PROG (PARI) {a(n) = if( n<0, 0, real( (10 + 7 * quadgen(8)) / 2 * (2 + quadgen(8))^n ))} /* Michael Somos, Sep 03 2013 */ (PARI) {a(n) = if( n<0, 0, n += 3 ; 2^ceil((n - 4)/2) * polcoeff( (1 + x - 3*x^2 - x^3) / (1 - 6*x^2 + x^4) + x * O(x^n), n))} /* Michael Somos, Sep 03 2013 */ (PARI) Vec((5-3*x)/(1-4*x+2*x^2) + O(x^30)) \\ Colin Barker, Dec 06 2015 CROSSREFS Cf. A079496, A143609. Sequence in context: A054113 A146697 A009229 * A180502 A261515 A171838 Adjacent sequences:  A010911 A010912 A010913 * A010915 A010916 A010917 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 17 05:23 EDT 2018. Contains 316275 sequences. (Running on oeis4.)