OFFSET
0,2
COMMENTS
Also 7*x^2 + 1 is a square; n=7 in PARI script below. - Cino Hilliard, Mar 08 2003
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
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
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
V. Thébault, Les Récréations Mathématiques. Gauthier-Villars, Paris, 1952, p. 281.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..200
H. Brocard, Notes élémentaires sur le problème de Peel [sic], Nouvelle Correspondance Mathématique, 4 (1878), 337-343.
M. Davis, One equation to rule them all, Trans. New York Acad. Sci. Ser. II, 30 (1968), 766-773.
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Mihai Prunescu, On other two representations of the C-recursive integer sequences by terms in modular arithmetic, arXiv:2406.06436 [math.NT], 2024. See p. 17.
Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 15.
Index entries for linear recurrences with constant coefficients, signature (16,-1).
FORMULA
G.f.: 3*x/(1-16*x+x^2).
From Mohamed Bouhamida, Sep 20 2006: (Start)
a(n) = 15*(a(n-1) + a(n-2)) - a(n-3).
a(n) = 17*(a(n-1) - a(n-2)) + a(n-3). (End)
a(n) = 16*a(n-1) - a(n-2) with a(1)=0 and a(2)=3. - Sture Sjöstedt, Nov 18 2011
E.g.f.: exp(8*x)*sinh(3*sqrt(7)*x)/sqrt(7). - G. C. Greubel, Feb 23 2019
MAPLE
A001080:=3*z/(1-16*z+z**2); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
MATHEMATICA
LinearRecurrence[{16, -1}, {0, 3}, 30] (* Harvey P. Dale, Nov 01 2011 *)
CoefficientList[Series[3*x/(1-16*x+x^2), {x, 0, 30}], x] (* G. C. Greubel, Dec 20 2017 *)
PROG
(PARI) nxsqp1(m, n) = { for(x=1, m, y = n*x*x+1; if(issquare(y), print1(x" ")) ) }
(PARI) x='x+O('x^30); concat([0], Vec(3*x/(1-16*x+x^2))) \\ G. C. Greubel, Dec 20 2017
(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
(SageMath) (3*x/(1-16*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 23 2019
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved