0,2

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 (blekraj(AT)yahoo.com), Jun 28 2002

a(n) solves for x in x^2 - 3*y^2 = 1, the corresponding y being given by A001353(n). 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 (blekraj(AT)yahoo.com), Sep 19 2003

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 (noe(AT)sspectra.com), May 04 2006

a(n)=2*a(n-1)+3*A001353(n-1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 21 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 (ck6(AT)evansville.edu), 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 (ck6(AT)evansville.edu), Aug 27 2008

a(n+1) is the Hankel transform of A000108(n)+A000984(n)=(n+2)*Catalan(n). [From Paul Barry (pbarry(AT)wit.ie), Aug 11 2009]

H. Brocard, Notes e'le'mentaires sur le proble`me de Peel, Nouvelles Correspondance Math\'{e}matique, 4 (1878), 161-169.

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

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

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'Interm\'{e}diaire des Math\'{e}maticiens, 10 (1903), 235-238.

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

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.

Chris Caldwell, Primality Proving, Arndt's theorem.

Tanya Khovanova, Recursive Sequences

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

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

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 (omomom(AT)hotmail.com), 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) = 2^(-n)*Sum_{k>=0} binomial(2n, 2k)*3^k = 2^(-n)*Sum_{k>=0} A086645(n, k)*3^k. - Philippe DELEHAM, 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 (pbarry(AT)wit.ie), 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. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 03 2004

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 (qntmpkt(AT)yahoo.com), Dec 27 2006

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

First differences of A001571. - njas, 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

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.

A001075:=-(-1+2*z)/(1-4*z+z**2); [S. Plouffe in his 1992 dissertation.]

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

(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

(Other) sage: [lucas_number2(n, 4, 1)/2 for n in xrange(0, 25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 14 2009]

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: A087448 A129273 A055988 this_sequence A113436 A126223 A114121

Adjacent sequences: A001072 A001073 A001074 this_sequence A001076 A001077 A001078

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 10 2000

Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 31 2002