A029547 Expansion of g.f. 1/(1 - 34*x + x^2).

1, 34, 1155, 39236, 1332869, 45278310, 1538129671, 52251130504, 1775000307465, 60297759323306, 2048348816684939, 69583562007964620, 2363792759454112141, 80299370259431848174, 2727814796061228725775, 92665403695822344828176, 3147895910861898495432209

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 and G. M. Gianella, On sums of squares of Pell-Lucas numbers, INTEGERS 6 (2006) #A15.
• R. Flórez, R. A. Higuita, and 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 sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (34,-1).

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).

Programs

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

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

Maple

with (combinat):seq(fibonacci(4*n+4,2)/12, n=0..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 *)

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

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

From Peter Bala, Dec 23 2012: (Start)

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

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

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