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A001080 a(n) = 16*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3.

%I A001080 M3155 N1278 #111 Feb 28 2025 09:40:22
%S A001080 0,3,48,765,12192,194307,3096720,49353213,786554688,12535521795,
%T A001080 199781794032,3183973182717,50743789129440,808716652888323,
%U A001080 12888722657083728,205410845860451325,3273684811110137472,52173546131901748227,831503053299317834160
%N A001080 a(n) = 16*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3.
%C A001080 Also 7*x^2 + 1 is a square; n=7 in PARI script below. - _Cino Hilliard_, Mar 08 2003
%C A001080 That is, the terms are solutions y of the Pell-Fermat equation x^2 - 7 * y^2 = 1. The corresponding values of x are in A001081. (x,y) = (1,0), (8,3), (127,48), ... - _Bernard Schott_, Feb 23 2019
%C A001080 The first solution to the equation x^2 - 7*y^2 = 1 is (X(0); Y(0)) = (1; 0) and the other solutions are defined by: (X(n); Y(n))= (8*X(n-1) + 21*Y(n-1); 3*X(n-1) + 8*Y(n-1)), with n >= 1. - _Mohamed Bouhamida_, Jan 16 2020
%D A001080 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001080 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001080 V. Thébault, Les Récréations Mathématiques. Gauthier-Villars, Paris, 1952, p. 281.
%H A001080 T. D. Noe, <a href="/A001080/b001080.txt">Table of n, a(n) for n = 0..200</a>
%H A001080 H. Brocard, <a href="http://resolver.sub.uni-goettingen.de/purl?PPN598948236_0004/DMDLOG_0053">Notes élémentaires sur le problème de Peel [sic]</a>, Nouvelle Correspondance Mathématique, 4 (1878), 337-343.
%H A001080 M. Davis, <a href="http://www.rand.org/pubs/research_memoranda/RM5494.html">One equation to rule them all</a>, Trans. New York Acad. Sci. Ser. II, 30 (1968), 766-773.
%H A001080 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A001080 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001080 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A001080 Mihai Prunescu, <a href="https://arxiv.org/abs/2406.06436">On other two representations of the C-recursive integer sequences by terms in modular arithmetic</a>, arXiv:2406.06436 [math.NT], 2024. See p. 17.
%H A001080 Mihai Prunescu and Lorenzo Sauras-Altuzarra, <a href="https://arxiv.org/abs/2405.04083">On the representation of C-recursive integer sequences by arithmetic terms</a>, arXiv:2405.04083 [math.LO], 2024. See p. 15.
%H A001080 Mihai Prunescu and Joseph M. Shunia, <a href="https://arxiv.org/abs/2502.16928">On modular representations of C-recursive integer sequences</a>, arXiv:2502.16928 [math.NT], 2025. See p. 5.
%H A001080 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-1).
%F A001080 G.f.: 3*x/(1-16*x+x^2).
%F A001080 From _Mohamed Bouhamida_, Sep 20 2006: (Start)
%F A001080 a(n) = 15*(a(n-1) + a(n-2)) - a(n-3).
%F A001080 a(n) = 17*(a(n-1) - a(n-2)) + a(n-3). (End)
%F A001080 a(n) = 16*a(n-1) - a(n-2) with a(1)=0 and a(2)=3. - _Sture Sjöstedt_, Nov 18 2011
%F A001080 E.g.f.: exp(8*x)*sinh(3*sqrt(7)*x)/sqrt(7). - _G. C. Greubel_, Feb 23 2019
%p A001080 A001080:=3*z/(1-16*z+z**2); # conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation
%t A001080 LinearRecurrence[{16,-1},{0,3},30] (* _Harvey P. Dale_, Nov 01 2011 *)
%t A001080 CoefficientList[Series[3*x/(1-16*x+x^2), {x, 0, 30}], x] (* _G. C. Greubel_, Dec 20 2017 *)
%o A001080 (PARI) nxsqp1(m,n) = { for(x=1,m, y = n*x*x+1; if(issquare(y),print1(x" ")) ) }
%o A001080 (PARI) x='x+O('x^30); concat([0], Vec(3*x/(1-16*x+x^2))) \\ _G. C. Greubel_, Dec 20 2017
%o A001080 (Magma) I:=[0,3]; [n le 2 select I[n] else 16*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 20 2017
%o A001080 (SageMath) (3*x/(1-16*x+x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 23 2019
%o A001080 (GAP) a:=[0,3];; for n in [3..30] do a[n]:=16*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Feb 23 2019
%Y A001080 Equals 3 * A077412. Bisection of A084069.
%Y A001080 Cf. A048907.
%Y A001080 Cf. A001081, A010727. - _Vincenzo Librandi_, Feb 16 2009
%K A001080 nonn,easy
%O A001080 0,2
%A A001080 _N. J. A. Sloane_