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A001075 a(0) = 1, a(1) = 2, a(n) = 4a(n-1) - a(n-2).
(Formerly M1769 N0700)
82
1, 2, 7, 26, 97, 362, 1351, 5042, 18817, 70226, 262087, 978122, 3650401, 13623482, 50843527, 189750626, 708158977, 2642885282, 9863382151, 36810643322, 137379191137, 512706121226, 1913445293767, 7141075053842, 26650854921601, 99462344632562, 371198523608647 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Chebyshev's T(n,x) polynomials evaluated at x=2.

x = 2^n - 1 is prime if and only if x divides a(2^(n-2)).

Any k in the sequence is succeeded by 2*k + sqrt{3*(k^2 - 1)} - Lekraj Beedassy, Jun 28 2002

This sequence gives the values of x in solutions of the Diophantine equation x^2 - 3*y^2 = 1; the corresponding values of y are in A001353. The solution ratios a(n)/A001353(n) are obtained as convergents of the continued fraction expansion of sqrt(3): either as successive convergents of [2;-4] or as odd convergents of [1;1,2]. - Lekraj Beedassy, Sep 19 2003 [edited by Jon E. Schoenfield, May 04 2014]

a(n) is half the central value in a list of three consecutive integers, the lengths of the sides of a triangle with integer sides and area. - Eugene McDonnell (eemcd(AT)mac.com), Oct 19 2003

a(3+6k)-1 and a(3+6k)+1 are consecutive odd powerful numbers. See A076445. - T. D. Noe, May 04 2006

The intermediate convergents to 3^(1/2), beginning with 3/2, 12/7, 45/26, 168/97, comprise a strictly increasing sequence; essentially, numerators=A005320, denominators=A001075. - Clark Kimberling, Aug 27 2008

The upper principal convergents to 3^(1/2), beginning with 2/1, 7/4, 26/15, 97/56, comprise a strictly decreasing sequence; numerators=A001075, denominators=A001353. - Clark Kimberling, Aug 27 2008

a(n+1) is the Hankel transform of A000108(n)+A000984(n)=(n+2)*Catalan(n). - Paul Barry, Aug 11 2009

Also, numbers such that floor[a(n)^2/3] is a square: base 3 analog of A031149, A204502, A204514, A204516, A204518, A204520, A004275, A001541. - M. F. Hasler, Jan 15 2012

Pisano period lengths:  1, 2, 2, 4, 3, 2, 8, 4, 6, 6, 10, 4, 12, 8, 6, 8, 18, 6, 5, 12,... - R. J. Mathar, Aug 10 2012

Except for the first term, positive values of x (or y) satisfying x^2 - 4xy + y^2 + 3 = 0. - Colin Barker, Feb 04 2014

Except for the first term, positive values of x (or y) satisfying x^2 - 14xy + y^2 + 48 = 0. - Colin Barker, Feb 10 2014

REFERENCES

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

Mcdonnell, Eugene, "Heron's Rule and Integer-Area Triangles", Vector 12.3 (January 1996) pp. 133-142

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).

P.-F. Teilhet, Reply to Query 2094, L'Intermédiaire des Mathématiciens, 10 (1903), 235-238.

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.

H. Brocard, Notes élémentaires sur le problème de Peel [sic], Nouvelle Correspondance Mathématique, 4 (1878), 337-343.

Chris Caldwell, Primality Proving, Arndt's theorem.

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.

Tanya Khovanova, Recursive Sequences

Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.

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.

F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83.

Index entries for sequences related to Chebyshev polynomials.

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients, signature (4,-1)

FORMULA

For all elements x of the sequence, 12*x^2 -12 is a square. Lim. as n-> Inf. a(n)/a(n-1) = 2 + sqrt(3) = (4 + sqrt(12))/2 which preserves the kinship with the equation "12*x^2 - 12 is a square" where the initial "12" ends up appearing as a square root. - Gregory V. Richardson, Oct 10 2002

a(n) = (S(n, 4) - S(n-2, 4))/2 = T(n, 2), with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. U, resp. T, are Chebyshev's polynomials of the second, resp. first, kind. S(n-1, 4) = A001353(n), n>=0. See A049310 and A053120.

a(n) = A001353(n+2)-2*A001353(n+1).

a(n) = 2^(-n)*Sum_{k>=0} binomial(2n, 2k)*3^k = 2^(-n)*Sum_{k>=0} A086645(n, k)*3^k. - Philippe Deléham, Mar 01, 2004

a(n) = ((2+sqrt(3))^n + (2-sqrt(3))^n)/2; a(n) = ceiling((1/2)*(2+sqrt(3))^(n)).

a(n) = cosh( n * ln( 2 + sqrt(3))).

a(n) = sum{k=0..floor(n/2); C(n, 2k)2^(n-2k)3^k }. - Paul Barry, May 08 2003

G.f.: (1-2x)/(1-4x+x^2). E.g.f.: exp(2x)cosh(sqrt(3)x). a(n)=4a(n-1)-a(n-2)=a(-n).

a(n+2) = 2*a(n+1) + 3*Sum_{k>=0} a(n-k)*2^k. - Philippe Deléham, Mar 03 2004

a(n) = 2*a(n-1)+3*A001353(n-1). - Lekraj Beedassy, Jul 21 2006

a(n) = left term of M^n * [1,0] where M = the 2 X 2 matrix [2,3; 1,2]. Right term = A001353(n). Example: a(4) = 97 since M^4 * [1,0] = [A001075(4), A001353(4)] = [97, 56]. - Gary W. Adamson, Dec 27 2006

Binomial transform of A026150: (1, 1, 4, 10, 28, 76,...). - Gary W. Adamson, Nov 23 2007

First differences of A001571. - N. J. A. Sloane, Nov 03 2009

Sequence satisfies -3 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 4*u*v. - Michael Somos, Sep 19 2008

a(n) = Sum_{k, 0<=k<=n} A201730(n,k)*2^k. - Philippe Deléham, Dec 06 2011

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k-4)/(x*(3*k-1) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013

a(n) = Sum_{k = 0..n} A238731(n,k). - Philippe Deléham, Mar 05 2014

EXAMPLE

2^6 -1 = 63 does not divide a(2^4) = 708158977, therefore 63 is composite. 2^5 -1 = 31 divides a(2^3) = 18817, therefore 31 is prime.

MAPLE

A001075:=-(-1+2*z)/(1-4*z+z**2); # Simon Plouffe in his 1992 dissertation

MATHEMATICA

Table[ Ceiling[(1/2)*(2 + Sqrt[3])^n], {n, 0, 24}]

f[x_] := (1-2*x) / (1-4*x+x^2); CoefficientList[ Series[ f[x], {x, 0, 24}], x] (* Jean-François Alcover, Dec 21 2011, after Simon Plouffe *)

a=1; b=1; Join[{1, 2}, Table[c=(b=b+a)+(a=a+b*2), {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)

PROG

(PARI) a(n)=subst(poltchebi(abs(n)), x, 2)

(PARI) a(n)=real((2+quadgen(12))^abs(n))

(PARI) a(n)=polsym(1-4*x+x^2, abs(n))[1+abs(n)]/2

(Sage) [lucas_number2(n, 4, 1)/2 for n in xrange(0, 25)] # Zerinvary Lajos, May 14 2009

(Haskell)

a001075 n = a001075_list !! n

a001075_list =

   1 : 2 : zipWith (-) (map (4 *) $ tail a001075_list) a001075_list

-- Reinhard Zumkeller, Aug 11 2011

(Sage)

def a(n):

    Q = QuadraticField(3, 't')

    u = Q.units()[0]

    return (u^n).lift().coeffs()[0]  # Ralf Stephan, Jun 19 2014

CROSSREFS

Cf. A065918, A071954. a(n) = sqrt(1+3*A001353(n)) (cf. Richardson comment).

Cf. A001353, A001571, A001834, A003500, A016064, A082840.

Bisections are A011943 and A094347.

Cf. A001353.

Cf. A026150.

Sequence in context: A188860 A129273 A055988 * A113436 A126223 A114121

Adjacent sequences:  A001072 A001073 A001074 * A001076 A001077 A001078

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, Jul 10 2000

Chebyshev comments from Wolfdieter Lang, Oct 31 2002

STATUS

approved

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Last modified December 22 06:09 EST 2014. Contains 252328 sequences.