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A144128 Chebyshev U(n,x) polynomial evaluated at x=18. 23
1, 36, 1295, 46584, 1675729, 60279660, 2168392031, 78001833456, 2805897612385, 100934312212404, 3630829342034159, 130608922001017320, 4698290362694589361, 169007844135004199676, 6079584098497456598975 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

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

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 *)

PROG

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

(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

(MAGMA) Z<x>:=PolynomialRing(Integers()); N<r>:=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

(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

(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

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).

Sequence in context: A170755 A218738 A158700 * A223405 A224267 A223924

Adjacent sequences:  A144125 A144126 A144127 * A144129 A144130 A144131

KEYWORD

nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

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

STATUS

approved

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Last modified April 10 16:16 EDT 2021. Contains 342845 sequences. (Running on oeis4.)