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A054320 G.f.: (1+x)/(1-10*x+x^2). 26
1, 11, 109, 1079, 10681, 105731, 1046629, 10360559, 102558961, 1015229051, 10049731549, 99482086439, 984771132841, 9748229241971, 96497521286869, 955226983626719, 9455772314980321, 93602496166176491, 926569189346784589 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Chebyshev's even-indexed U-polynomials evaluated at sqrt(3).

a(n)^2 is a star number (A003154).

a(n) = L(n,-10)*(-1)^n, where L is defined as in A108299; see also A072256 for L(n,+10). - Reinhard Zumkeller, Jun 01 2005

(sqrt(2)+sqrt(3))^(2*n+1)=a(n)*sqrt(2)+A138288(n)*sqrt(3); a(n)=A138288(n)+A001078(n). - Reinhard Zumkeller, Mar 12 2008

a(n) give the values of x solving: 3y^2 - 2x^2 = 1. Corresponding values of y are given by A072256(n+1). x + y = A001078(n+1). - Richard R. Forberg, Nov 21 2013

The aerated sequence (b(n))n>=1 = [1, 0, 9, 0, 71, 0, 559, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -8, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047. - Peter Bala, Mar 22 2015

LINKS

Table of n, a(n) for n=0..18.

Tanya Khovanova, Recursive Sequences

Eric Weisstein's World of Mathematics, Star Number

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume

Index entries for linear recurrences with constant coefficients, signature (10,-1).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

(a(n)-1)^2+a(n)^2+(a(n)+1)^2=b(n)^2+(b(n)+1)^2=c(n), where b(n) is A031138 and c(n) is A007667

Any k in the sequence has the successor 5*k + 2sqrt{3(2*k^2 + 1)}. - Lekraj Beedassy, Jul 08 2002

a(n) = 10*a(n-1) - a(n-2); a(n)=(sqrt(6) - 2)/4*(5 + 2*sqrt(6))^n - (sqrt(6) + 2)/4*(5 - 2*sqrt(6))^n.

a(n) = U(2*(n-1), sqrt(3)) = S(n-1, 10) + S(n-2, 10) with Chebyshev's U(n, x) and S(n, x) := U(n, x/2) polynomials and S(-1, x) := 0. S(n, 10) = A004189(n+1), n>=0.

For all members x of the sequence, 6*x^2 + 3 is a square. Lim. n-> Inf. a(n)/a(n-1) = 5 + 2*sqrt(6) - Gregory V. Richardson, Oct 13 2002

a(n) = [ [(5+2*sqrt(6))^n - (5-2*sqrt(6))^n] + [(5+2*sqrt(6))^(n-1) - (5-2*sqrt(6))^(n-1)] / (4*sqrt(6)) - Gregory V. Richardson, Oct 13 2002

Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n, -12)=a(n) - Benoit Cloitre, Nov 10 2002

a(n) = A001079(n) + 3*A001078(n). - Reinhard Zumkeller, Mar 12 2008

A054320(n) = A142238(2n) = A041006(2n)/2 = A041038(2n)/4 [From M. F. Hasler, Feb 14 2009]

a(n) = sqrt(A006061(n)). - Zak Seidov, Oct 22 2012

a(n) = sqrt((3* A072256(n)^2 - 1)/2).

EXAMPLE

a(1)^2=121 is the 5th star number (A003154).

MATHEMATICA

q=12; s=0; lst={}; Do[s+=n; If[Sqrt[q*s+1]==Floor[Sqrt[q*s+1]], AppendTo[lst, Sqrt[q*s+1]]], {n, 0, 9!}]; lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)

CoefficientList[Series[(1 + x) / (1 - 10 x + x^2), {x, 0, 33}], x] (* Vincenzo Librandi, Mar 22 2015 *)

a[c_, n_] := Module[{},

   p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];

   d := Numerator[Convergents[Sqrt[c], n p]];

   t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];

   Return[t];

] (* Complement of A142238 *)

a[3/2, 20] (* Gerry Martens, Jun 07 2015 *)

PROG

(PARI) a(n)=if(n<1, 0, subst(poltchebi(n)-poltchebi(n-1), x, 5)/4)

(Sage) [(lucas_number2(n, 10, 1)-lucas_number2(n-1, 10, 1))/8 for n in xrange(1, 20)]# [Zerinvary Lajos, Nov 10 2009]

(MAGMA) I:=[1, 11]; [n le 2 select I[n] else 10*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 22 2015

CROSSREFS

A member of the family A057078, A057077, A057079, A005408, A002878, A001834, A030221, A002315, A033890, A057080, A057081, A054320, which are the expansions of (1+x) / (1-kx+x^2) with k = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. - Philippe Deléham, May 04 2004

Cf. A003154, A006061, A031138, A007667, A004189.

Cf. A138281. Cf. A100047.

Cf. A142238.

Sequence in context: A125423 A165149 A048346 * A124290 A094703 A169631

Adjacent sequences:  A054317 A054318 A054319 * A054321 A054322 A054323

KEYWORD

easy,nonn

AUTHOR

Ignacio Larrosa Cañestro, Feb 27 2000

EXTENSIONS

Chebyshev comments from Wolfdieter Lang, Oct 31 2002

STATUS

approved

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Last modified March 30 16:27 EDT 2017. Contains 284302 sequences.