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%I #90 Sep 02 2023 19:22:03
%S 2,2,3,6,14,35,90,234,611,1598,4182,10947,28658,75026,196419,514230,
%T 1346270,3524579,9227466,24157818,63245987,165580142,433494438,
%U 1134903171,2971215074,7778742050,20365011075,53316291174,139583862446,365435296163,956722026042
%N One of four 3rd-order recurring sequences for which the first derived sequence and the Galois transformed sequence coincide.
%C a(n) is also a sequence with the property that the difference between the sum and product of two consecutive terms is equal to the square of the difference between those terms, i.e., a(n)*a(n+1) - (a(n)+ a(n+1)) = (a(n) - a(n + 1))^2. The difference between those two terms, a(n + 1) - a(n) = F(2n -2), the (2n - 2)th Fibonacci number. - _John Baker_, May 18 2010
%C Conjecture: consecutive terms of this sequence and consecutive terms of A101265 provide all the positive integer solutions of (a+b)*(a+b+1) == 0 (mod (a*b)). - _Robert Israel_, Aug 26 2015
%C Conjecture is true: see Mathematics Stack Exchange link. - _Robert Israel_, Sep 06 2015
%C Consecutive terms of this sequence and consecutive terms of A101879 provide all the positive integer pairs for which K = (a+1)/b + (b+1)/a is integer. For this sequence, K = 3. - _Andrey Vyshnevyy_, Sep 18 2015
%D L. E. Dickson, History of the Theory of Numbers, Dover, New York, 1971.
%H Robert Israel, <a href="/A032908/b032908.txt">Table of n, a(n) for n = 0..2154</a>
%H M. Elia, <a href="http://dx.doi.org/10.1007/978-94-011-5020-0_12">A Note on derived linear recurring sequences</a>, pp. 83-92 of Proceedings Seventh Int. Conference on Fibonacci Numbers and their Applications (Austria, 1996), Applications of Fibonacci Numbers, Volume 7.
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=919">Encyclopedia of Combinatorial Structures 919</a>.
%H R. Israel, W. Jagy et al., <a href="http://math.stackexchange.com/questions/1417404/diophantine-equation-xyxy1-kxy-0">Diophantine equation (x+y)(x+y+1)-kxy=0</a>, Mathematics Stack Exchange, Sep 1 2015.
%H H. D. Nguyen and D. Taggart, <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.391.2522&rep=rep1&type=pdf">Mining the OEIS: Ten Experimental Conjectures</a>, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).
%F G.f.: (2 - 6*x + 3*x^2)/((1 - x)*(1 - 3*x + x^2)).
%F a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
%F a(n) = Fibonacci(2*n-1) + 1 = A001519(n) + 1. - _Vladeta Jovovic_, Mar 19 2003
%F a(n) = 3*a(n - 1) - a(n - 2) - 1. - _N. Sato_, Jan 21 2010
%F From _Wolfdieter Lang_, Aug 27 2014: (Start)
%F a(n) = 1 + S(n-1, 3) - S(n-2, 3) = 1 + A001519(n), with Chebyshev S-polynomials (see A049310). For n < 0, we have S(-1, x) = 0 and S(-2, x) = -1.
%F This follows from the partial fraction decomposition of the g.f., 1/(1 - x) + (1 - 2*x)/ (1 - 3*x + x^2), using the recurrence for S, or from A001519. (End)
%F From _Robert Israel_, Aug 26 2015: (Start)
%F (a(n) + a(n+1))*(a(n) + a(n+1) + 1) = 5*a(n)*a(n+1).
%F a(n+1) = (3*a(n) + sqrt(5*a(n)^2 - 10*a(n) + 1) - 1)/2 for n >= 1. (End)
%F a(n) = 1 + (2^(-1-n) * ((3 - sqrt(5))^n * (1 + sqrt(5)) + (-1 + sqrt(5)) * (3 + sqrt(5))^n)) / sqrt(5). - _Colin Barker_, Nov 02 2016
%F Sum_{n>=0} 1/a(n) = phi (A001622). - _Amiram Eldar_, Oct 05 2020
%p f:= proc(n) option remember; local x;
%p x:= procname(n-1); (3*x + sqrt(5*x^2 - 10*x + 1) - 1)/2 end proc:
%p f(0):= 2: f(1):= 2:
%p map(f, [$0..30]); # _Robert Israel_, Aug 26 2015
%t Table[Fibonacci[2 n - 1] + 1, {n, 0, 27}] (* _Michael De Vlieger_, Aug 26 2015 *)
%t LinearRecurrence[{4,-4,1},{2,2,3},40] (* _Harvey P. Dale_, Apr 11 2018 *)
%o (PARI) Vec((2-6*x+3*x^2)/(1-4*x+4*x^2-x^3)+O(x^66)) \\ _Joerg Arndt_, Jul 02 2013
%o (Magma) [2] cat [n le 1 select 2 else Floor((3*Self(n-1) + Sqrt(5*Self(n-1)^2 - 10*Self(n-1) + 1) - 1)/2): n in [1..30]]; // _Vincenzo Librandi_, Aug 27 2015
%Y Cf. A001519, A001622, A049310, A101265.
%K eigen,nonn,easy
%O 0,1
%A Michele Elia (elia(AT)polito.it)
%E More terms from _Ralf Stephan_, Mar 10 2003
%E Index for Chebyshev polynomials and cross reference added by _Wolfdieter Lang_, Aug 27 2014
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