A144128 Chebyshev U(n,x) polynomial evaluated at x=18.

1, 36, 1295, 46584, 1675729, 60279660, 2168392031, 78001833456, 2805897612385, 100934312212404, 3630829342034159, 130608922001017320, 4698290362694589361, 169007844135004199676, 6079584098497456598975

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

Keywords

• nonn
• easy

Comments

From Bruno Berselli, Nov 21 2011: (Start)

A Diophantine property of these numbers: ((a(n+1)-a(n-1))/2)^2 - 323*a(n)^2 = 1.

More generally, for t(m) = m + sqrt(m^2-1) and u(n) = (t(m)^n - 1/t(m)^n)/(t(m) - 1/t(m)), we can verify that ((u(n+1) - u(n-1))/2)^2 - (m^2-1)*u(n)^2 = 1. (End)

a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,35}. - Milan Janjic, Jan 26 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Index entries for linear recurrences with constant coefficients, signature (36,-1).

Crossrefs

Cf. A200441, A200442, A200724 (incomplete list).

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), this sequence (m=18), A078987 (m=19), A097316 (m=33).

Programs

GAP

a:=[1,36];; for n in [3..20] do a[n]:=36*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Feb 09 2018

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-323); S:=[((18+r)^n-1/(18+r)^n)/(2*r): n in [1..15]]; [Integers()!S[j]: j in [1..#S]]; // Bruno Berselli, Nov 21 2011

Magma

I:=[1,36]; [n le 2 select I[n] else 36*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011

Maple

seq( simplify(ChebyshevU(n, 18)), n=0..20); # G. C. Greubel, Dec 22 2019

Mathematica

LinearRecurrence[{36,-1},{1,36},20] (* Vincenzo Librandi, Nov 22 2011 *)
GegenbauerC[Range[0,20],1,18] (* Harvey P. Dale, May 19 2019 *)
ChebyshevU[Range[21] -1, 18] (* G. C. Greubel, Dec 22 2019 *)

Maxima

makelist(sum((-1)^k*binomial(n-1-k,k)*36^(n-1-2*k),k,0,floor(n/2)),n,1,15); /* Bruno Berselli, Nov 21 2011 */

PARI

Vec(x/(1-36*x+x^2)+O(x^16)) \\ Bruno Berselli, Nov 21 2011

PARI

a(n) = polchebyshev(n, 2, 18); \\ Michel Marcus, Feb 09 2018

Sage

[lucas_number1(n,36,1) for n in range(1, 16)] # Zerinvary Lajos, Nov 07 2009

Formula

From Bruno Berselli, Nov 21 2011: (Start)

G.f.: x/(1-36*+x^2).

a(n) = 36*a(n-1) - a(n-2) with a(1)=1, a(2)=36.

a(n) = (t^n - 1/t^n)/(t - 1/t) for t = 18+sqrt(323).

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-1-k, k)*36^(n-1-2*k). (End)

a(n) = Sum_{k=0..n} A101950(n,k)*35^k. - Philippe Deléham, Feb 10 2012

Product {n >= 1} (1 + 1/a(n)) = 1/17*(17 + sqrt(323)). - Peter Bala, Dec 23 2012

Product {n >= 2} (1 - 1/a(n)) = 1/36*(17 + sqrt(323)). - Peter Bala, Dec 23 2012

Extensions

As Michel Marcus points out, some parts of this entry assume the offset is 1, others parts assume the offset is 0. The whole entry needs careful editing. - N. J. A. Sloane, Feb 10 2018