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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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30
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of units of a(n) belongs to a periodic sequence: 0, 1, 8, 3, 6, 5, 4, 7, 2, 9. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009]
a(n) solves for y in the Diophantine equation x^2-15*y^2=1, The corresponding x solutions are provided by A001091.[From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2010]
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). [From John M. Campbell, Jul 08 2011]
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REFERENCES
| H. Brocard, Notes e'le'mentaires sur le proble`me de Peel, Nouvelles Correspondance Math\'{e}matique, 4 (1878), 161-169.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pps. 231-242.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=8, q=-1.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=10.
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
| T. D. Noe, Table of n, a(n) for n=0..100
Tanya Khovanova, Recursive Sequences
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients, signature (8,-1).
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FORMULA
| 15*a(n)^2 - A001091(n)^2 = -1.
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)=x/(1-8x+x^2). - Barry E. Williams, Aug 18 2000
Lim. n-> Inf. a(n)/a(n-1) = 4 + sqrt(15). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 13 2002
a(n) = 7*(a(n-1)+a(n-2))-a(n-3). a(n) = 9*(a(n-1)-a(n-2))+a(n-3). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Feb 07 2007
[A070997(n-1), a(n)] = [1,6; 1,7]^n * [1,0]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 21 2008
a(-n) = -a(n). - Michael Somos Apr 05 2008
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*7^k. - DELEHAM Philippe, Feb 10 2012
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EXAMPLE
| x + 8*x^2 + 63*x^3 + 496*x^4 + 3905*x^5 + 30744*x^6 + 242047*x^7 + ...
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MAPLE
| A001090:=1/(1-8*z+z**2); [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 4]], {n, 0, 6^2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 11 2008]
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PROG
| (PARI) {a(n) = subst(poltchebi(n+1) - 4 * poltchebi(n), x, 4) / 15} /* Michael Somos Apr 05 2008 */
sage: [lucas_number1(n, 8, 1) for n in range(22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
(Other) sage: [lucas_number1(n, 8, 1) for n in xrange(0, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
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CROSSREFS
| Cf. A000027, A001906, A001353, A004254, A001109, A004187, A001091.
a(n)=sqrt((A001091(n)^2-1)/15).
Cf. A070997.
Sequence in context: A171313 A081107 A164592 * A105219 A060071 A037205
Adjacent sequences: A001087 A001088 A001089 * A001091 A001092 A001093
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KEYWORD
| nonn,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 02 2000
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