A001080 a(n) = 16*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3. (Formerly M3155 N1278)

0, 3, 48, 765, 12192, 194307, 3096720, 49353213, 786554688, 12535521795, 199781794032, 3183973182717, 50743789129440, 808716652888323, 12888722657083728, 205410845860451325, 3273684811110137472, 52173546131901748227, 831503053299317834160

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

Equals 3 * A077412. Bisection of A084069.

Cf. A048907.

Cf. A001081, A010727. - Vincenzo Librandi, Feb 16 2009

Sequence in context: A264730 A024042 A007654 * A099852 A270005 A218382

Adjacent sequences: A001077 A001078 A001079 * A001081 A001082 A001083

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved