%I #74 Apr 16 2023 20:36:28

%S 1,34,1155,39236,1332869,45278310,1538129671,52251130504,

%T 1775000307465,60297759323306,2048348816684939,69583562007964620,

%U 2363792759454112141,80299370259431848174,2727814796061228725775,92665403695822344828176,3147895910861898495432209

%N Expansion of g.f. 1/(1 - 34*x + x^2).

%C Chebyshev sequence U(n,17)=S(n,34) with Diophantine property.

%C b(n)^2 - 2*(12*a(n))^2 = 1 with the companion sequence b(n)=A056771(n+1). - _Wolfdieter Lang_, Dec 11 2002

%C 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

%C a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,33}. - _Milan Janjic_, Jan 26 2015

%H Indranil Ghosh, <a href="/A029547/b029547.txt">Table of n, a(n) for n = 0..651</a> (terms 0..200 from Vincenzo Librandi)

%H Z. Cerin and G. M. Gianella, <a href="https://www.emis.de/journals/INTEGERS/papers/g15/g15.Abstract.html">On sums of squares of Pell-Lucas numbers</a>, INTEGERS 6 (2006) #A15.

%H R. Flórez, R. A. Higuita, and A. Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mukherjee/mukh2.html">Alternating Sums in the Hosoya Polynomial Triangle</a>, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (34,-1).

%F a(n) = 34*a(n-1) - a(n-2), a(-1)=0, a(0)=1.

%F 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

%F a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap = 17+12*sqrt(2) and am = 17-12*sqrt(2).

%F a(n) = Sum_{k = 0..floor(n/2)} (-1)^k*binomial(n-k, k)*34^(n-2*k).

%F a(n) = sqrt((A056771(n+1)^2 -1)/2)/12.

%F 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

%F a(n) = Sum_{k=0..n} A101950(n,k)*33^k. - _Philippe Deléham_, Feb 10 2012

%F From _Peter Bala_, Dec 23 2012: (Start)

%F Product {n >= 0} (1 + 1/a(n)) = 1/4*(4 + 3*sqrt(2)).

%F Product {n >= 1} (1 - 1/a(n)) = 2/17*(4 + 3*sqrt(2)). (End)

%F E.g.f.: exp(17*x)*(24*cosh(12*sqrt(2)*x) + 17*sqrt(2)*sinh(12*sqrt(2)*x))/24. - _Stefano Spezia_, Apr 16 2023

%p with (combinat):seq(fibonacci(4*n+4,2)/12, n=0..15); # _Zerinvary Lajos_, Apr 21 2008

%t Table[GegenbauerC[n, 1, 17], {n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008 *)

%t LinearRecurrence[{34,-1},{1,34},20] (* _Vincenzo Librandi_, Nov 22 2011 *)

%t ChebyshevU[Range[21] -1, 17] (* _G. C. Greubel_, Dec 22 2019 *)

%o (PARI) A029547(n, x=[0,1],A=[17,72*4;1,17]) = vector(n,i,(x*=A)[1]) \\ _M. F. Hasler_, May 26 2007

%o (PARI) vector( 21, n, polchebyshev(n-1, 2, 17) ) \\ _G. C. Greubel_, Dec 22 2019

%o (Sage) [lucas_number1(n,34,1) for n in range(1, 16)] # _Zerinvary Lajos_, Nov 07 2009

%o (Sage) [chebyshev_U(n,17) for n in (0..20)] # _G. C. Greubel_, Dec 22 2019

%o (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

%o (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

%Y A091761 is an essentially identical sequence.

%Y Cf. A200441, A200724.

%Y 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).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_