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A001090
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a(n) = 8*a(n-1) - a(n-2); a(0) = 0, a(1) = 1.
(Formerly M4554 N1936)
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52
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0, 1, 8, 63, 496, 3905, 30744, 242047, 1905632, 15003009, 118118440, 929944511, 7321437648, 57641556673, 453811015736, 3572846569215, 28128961537984, 221458845734657, 1743541804339272, 13726875588979519, 108071462907496880, 850844827670995521, 6698687158460467288
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OFFSET
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0,3
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COMMENTS
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This sequence gives the values of y in solutions of the Diophantine equation x^2 - 15*y^2 = 1; the corresponding values of x are in A001091. - Vincenzo Librandi, Nov 12 2010 [edited by Jon E. Schoenfield, May 02 2014]
For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 8's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n >= 1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,7}. - Milan Janjic, Jan 25 2015
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REFERENCES
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Julio R. Bastida, Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009) - From N. J. A. Sloane, May 30 2012
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = sqrt((A001091(n)^2 - 1)/15).
a(n) = S(2*n-1, sqrt(10))/sqrt(10) = S(n-1, 8); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310, with S(-1, x) := 0.
a(n) = ((4+sqrt(15))^n - (4-sqrt(15))^n)/(2*sqrt(15)).
G.f.: x/(1-8*x+x^2). (End)
Product_{n >= 1} (1 + 1/a(n)) = (1/3)*(3 + sqrt(15)).
Product_{n >= 2} (1 - 1/a(n)) = (1/8)*(3 + sqrt(15)).
(End)
E.g.f.: exp(4*x)*sinh(sqrt(15)*x)/sqrt(15). - Stefano Spezia, Dec 12 2022
a(n) = Sum_{k = 0..n-1} binomial(n+k, 2*k+1)*6^k = Sum_{k = 0..n-1} (-1)^(n+k+1)* binomial(n+k, 2*k+1)*10^k. - Peter Bala, Jul 17 2023
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EXAMPLE
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G.f. = x + 8*x^2 + 63*x^3 + 496*x^4 + 3905*x^5 + 30744*x^6 + 242047*x^7 + ...
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MAPLE
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seq( simplify(ChebyshevU(n-1, 4)), n=0..20); # G. C. Greubel, Dec 23 2019
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MATHEMATICA
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LinearRecurrence[{8, -1}, {0, 1}, 30] (* Harvey P. Dale, Aug 29 2012 *)
CoefficientList[Series[x/(1-8*x+x^2), {x, 0, 20}], x] (* G. C. Greubel, Dec 20 2017 *)
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PROG
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(PARI) {a(n) = subst(poltchebi(n+1) - 4 * poltchebi(n), x, 4) / 15}; /* Michael Somos, Apr 05 2008 */
(PARI) {a(n) = polchebyshev(n-1, 2, 4)}; /* Michael Somos, May 28 2014 */
(PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1-8*x-x^2))) \\ G. C. Greubel, Dec 20 2017
(SageMath) [lucas_number1(n, 8, 1) for n in range(22)] # Zerinvary Lajos, Jun 25 2008
(SageMath) [chebyshev_U(n-1, 4) for n in (0..20)] # G. C. Greubel, Dec 23 2019
(Magma) I:=[0, 1]; [n le 2 select I[n] else 8*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 20 2017
(GAP) m:=4;; a:=[0, 1];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 23 2019
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CROSSREFS
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Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), this sequence (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).
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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