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A001080 a(n) = 16*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3.
(Formerly M3155 N1278)
7
0, 3, 48, 765, 12192, 194307, 3096720, 49353213, 786554688, 12535521795, 199781794032, 3183973182717, 50743789129440, 808716652888323, 12888722657083728, 205410845860451325, 3273684811110137472, 52173546131901748227, 831503053299317834160 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also 7*x^2+1 is a square. n=7 in PARI script below. - Cino Hilliard, Mar 08 2003

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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (16,-1)

FORMULA

G.f.: 3*x/(1-16*x+x^2).

From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), 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)=(1/14)*sqrt(7)*[8+3*sqrt(7)]^n-(1/14)*[8-3*sqrt(7)]^n*sqrt(7), with n>=0. - Paolo P. Lava, Oct 02 2008

a(n) = 16*a(n-1) - a(n-2) with a(1)=0 and a(2)=3. - Sture Sjöstedt, Nov 18 2011

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, 50}], 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); 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 *)

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

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Last modified October 16 05:44 EDT 2018. Contains 316259 sequences. (Running on oeis4.)