The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A029547 Expansion of 1/(1-34*x+x^2). 31
 1, 34, 1155, 39236, 1332869, 45278310, 1538129671, 52251130504, 1775000307465, 60297759323306, 2048348816684939, 69583562007964620, 2363792759454112141, 80299370259431848174, 2727814796061228725775 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Chebyshev sequence U(n,17)=S(n,34) with Diophantine property. b(n)^2 - 2*(12*a(n))^2 = 1 with the companion sequence b(n)=A056771(n+1). - Wolfdieter Lang, Dec 11 2002 More generally, for t(m) = m + sqrt(m^2-1) and u(n) = (t(m)^(n+1) - 1/t(m)^(n+1))/(t(m) - 1/t(m)), we can verify that ((u(n+1) - u(n-1))/2)^2 - (m^2-1)*u(n)^2 = 1. - Bruno Berselli, Nov 21 2011 a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,33}. - Milan Janjic, Jan 26 2015 LINKS Indranil Ghosh, Table of n, a(n) for n = 0..651 (terms 0..200 from Vincenzo Librandi) Z. Cerin, G. M. Gianella, On sums of squares of Pell-Lucas numbers, INTEGERS 6 (2006) #A15. R. Flórez, R. A. Higuita, A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014). Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (34,-1). FORMULA a(n) = 34*a(n-1) - a(n-2), a(-1)=0, a(0)=1. a(n) = S(n, 34) with S(n, x):= U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310. - Wolfdieter Lang, Dec 11 2002 a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap = 17+12*sqrt(2) and am = 17-12*sqrt(2). a(n) = Sum_{k = 0..floor(n/2)} (-1)^k*binomial(n-k, k)*34^(n-2*k). a(n) = sqrt((A056771(n+1)^2 -1)/2)/12. a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) with a(-1)=0, a(0)=1, a(1)=34. Also a(n) = (sqrt(2)/48)*((17+12*sqrt(2))^n-(17-12*sqrt(2))^n) = (sqrt(2)/48)*((3+2*sqrt(2))^(2n+2)-(3-2*sqrt(2))^(2n+2)) = (sqrt(2)/48)*((1+sqrt(2))^(4n+4)-(1-sqrt(2))^(4n+4)). - Antonio Alberto Olivares, Mar 19 2008 a(n) = Sum_{k=0..n} A101950(n,k)*33^k. - Philippe Deléham, Feb 10 2012 Product {n >= 0} (1 + 1/a(n)) = 1/4*(4 + 3*sqrt(2)). - Peter Bala, Dec 23 2012 Product {n >= 1} (1 - 1/a(n)) = 2/17*(4 + 3*sqrt(2)). - Peter Bala, Dec 23 2012 MAPLE with (combinat):seq(fibonacci(4*n, 2)/12, n=1..15); # Zerinvary Lajos, Apr 21 2008 MATHEMATICA Table[GegenbauerC[n, 1, 17], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *) LinearRecurrence[{34, -1}, {1, 34}, 20] (* Vincenzo Librandi, Nov 22 2011 *) ChebyshevU[Range[21] -1, 17] (* G. C. Greubel, Dec 22 2019 *) PROG (PARI) A029547(n, x=[0, 1], A=[17, 72*4; 1, 17]) = vector(n, i, (x*=A)[1]) \\ M. F. Hasler, May 26 2007 (PARI) vector( 21, n, polchebyshev(n-1, 2, 17) ) \\ G. C. Greubel, Dec 22 2019 (Sage) [lucas_number1(n, 34, 1) for n in range(1, 16)] # Zerinvary Lajos, Nov 07 2009 (Sage) [chebyshev_U(n, 17) for n in (0..20)] # G. C. Greubel, Dec 22 2019 (MAGMA) I:=[1, 34]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011 (GAP) m:=17;; a:=[1, 2*m];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019 CROSSREFS A091761 is an essentially identical sequence. Cf. A200441, A200724. Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33). Sequence in context: A218736 A248163 A158696 * A091761 A264134 A264019 Adjacent sequences:  A029544 A029545 A029546 * A029548 A029549 A029550 KEYWORD nonn,easy AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 25 16:34 EST 2020. Contains 338625 sequences. (Running on oeis4.)