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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.
• Mihai Prunescu and Joseph M. Shunia, On modular representations of C-recursive integer sequences, arXiv:2502.16928 [math.NT], 2025. See p. 5.
• Index entries for linear recurrences with constant coefficients, signature (16,-1).

Crossrefs

Equals 3 * A077412. Bisection of A084069.

Cf. A048907.

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

Programs

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

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

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 *)

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

SageMath

(3*x/(1-16*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 23 2019

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