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A029548
Expansion of 1/(1 - 32*x + x^2).
22
1, 32, 1023, 32704, 1045505, 33423456, 1068505087, 34158739328, 1092011153409, 34910198169760, 1116034330278911, 35678188370755392, 1140585993533893633, 36463073604713840864, 1165677769357309014015
OFFSET
0,2
COMMENTS
From Bruno Berselli, Nov 21 2011: (Start)
A Diophantine property of these numbers: ((a(n+1) - a(n-1))/2)^2 - 255*a(n)^2 = 1.
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. (End)
a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,31}. - Milan Janjic, Jan 26 2015
FORMULA
a(n) = 32*a(n-1) - a(n-2), a(-1)=0, a(0)=1.
a(n) = S(n, 32) with S(n, x) = U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310. - Wolfdieter Lang, Nov 29 2002
a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap=16+sqrt(255) and am=16-sqrt(255).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*32^(n-2*k).
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*31^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/15*(15 + sqrt(255)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/32*(15 + sqrt(255)). - Peter Bala, Dec 23 2012
MAPLE
seq(simplify(ChebyshevU(n, 16)), n=0..20); # G. C. Greubel, Dec 22 2019
MATHEMATICA
Table[GegenbauerC[n, 1, 16], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
CoefficientList[Series[1/(1-32x+x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Dec 24 2012 *)
ChebyshevU[Range[0, 20], 16] (* G. C. Greubel, Dec 22 2019 *)
PROG
(SageMath) [lucas_number1(n, 32, 1) for n in range(1, 16)] # Zerinvary Lajos, Nov 07 2009
(SageMath) [chebyshev_U(n, 16) for n in (0..20)] # G. C. Greubel, Dec 22 2019
(Magma) I:=[1, 32]; [n le 2 select I[n] else 32*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012
(PARI) vector( 21, n, polchebyshev(n-1, 2, 17) ) \\ G. C. Greubel, Dec 22 2019
(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
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), this sequence (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33), A097725 (m=51).
Sequence in context: A300265 A158617 A171337 * A016745 A189267 A223672
KEYWORD
nonn,easy
STATUS
approved