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
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).
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