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 A054320 Expansion of g.f.: (1+x)/(1-10*x+x^2). 28
 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 From Reinhard Zumkeller, Mar 12 2008: (Start) (sqrt(2) + sqrt(3))^(2*n+1) = a(n)*sqrt(2) + A138288(n)*sqrt(3); a(n) = A138288(n) + A001078(n). (End) {a(n)} give the values of x solving: 3*y^2 - 2*x^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 REFERENCES Fink, Alex, Richard Guy, and Mark Krusemeyer. "Partitions with parts occurring at most thrice." Contributions to Discrete Mathematics 3.2 (2008), 76-114. See Section 13. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9. 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). 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 + 2*sqrt(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->infinity} 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. - 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). - T. D. Noe, Oct 23 2012 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)/(1-10x+x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 22 2015 *) a[c_, n_] := Module[{},    p := Length[ContinuedFraction[ Sqrt[ c]][]];    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, 30)] # 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 (GAP) a:=[1, 11];; for n in [3..30] do a[n]:=10*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jul 22 2019 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 * A287836 A124290 A094703 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 October 18 20:27 EDT 2019. Contains 328197 sequences. (Running on oeis4.)