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A087130
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a(n)=5*a(n-1)+a(n-2); a(0)=2, a(1)=5.
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10
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2, 5, 27, 140, 727, 3775, 19602, 101785, 528527, 2744420, 14250627, 73997555, 384238402, 1995189565, 10360186227, 53796120700, 279340789727, 1450500069335, 7531841136402, 39109705751345, 203080369893127
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Sequence related to the 'silver mean' [5;5,5,5,5,...].
The solution to the general recurrence a(n)=(2k+1)a(n-1)+a(n-2), a(0)=2, a(1)=2k+1 is a(n)=((2k+1)+sqrt(4k^2+4k+5))^n+(2k+1)-sqrt(4k^2+4k+5))^n)/2; a(n)=2^(1-n)sum{j=0..n, C(n, 2j)(4k^2+4k+5)^j(2k+1)^(n-2j)}; a(n)=2T(n, (2k+1)x/2)(-1)^i with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003
Primes in this sequence include a(0) = 2; a(1) = 5; a(4) = 727; a(8) = 528527 (3) semiprimes in this sequence include a(7) = 101785; a(13) = 1995189565; a(16) = 279340789727; a(19) = 39109705751345; a(20) = 203080369893127 - Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 09 2005
[a(n)]^2 - 29*(A052918(n-1)^2 = 4*(-1)^n; n>0 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 07 2008]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 12 2010: (Start)
For more information about this type of recurrence follow the Khovanova link and see A054413 and A086902.
(End)
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 01 2010: (Start)
Binomial transform of A072263.
(End)
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n)=((5+sqrt(29))/2)^n+((5-sqrt(29))/2)^n
E.g.f. : 2exp(5x/2)cosh(sqrt(29)x/2); a(n)=2^(1-n)sum{k=0..floor(n/2), C(n, 2k)29^k5^(n-2k)}. a(n)=2T(n, 5i/2)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003
O.g.f.: (-2+5*x)/(-1+5*x+x^2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 02 2007
a(-n) = (-1)^n * a(n). - Michael Somos Nov 01 2008
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 12 2010: (Start)
Limit(a(n+k)/a(k), k=infinity) = (A087130(n) + A052918(n-1)*sqrt(29))/2.
Limit(A087130(n)/A052918(n-1), n=infinity) = sqrt(29).
(End)
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PROG
| (PARI) {a(n) = if( n<0, (-1)^n * a(-n), polsym(x^2 - 5*x -1, n) [n + 1])} /* Michael Somos Nov 04 2008 */
(Other) sage: [lucas_number2(n, 5, -1) for n in xrange(0, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 14 2009]
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CROSSREFS
| Cf. A006497, A014448, A085447.
Cf. A086902, A000032.
Equals A100236(n) + 1.
A052918 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 07 2008]
A090248(n) = a(2*n). 5 * A097834(n) = a(2*n + 1). - Michael Somos Nov 01 2008
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 12 2010: (Start)
a(3n+1) = A041046(5n), a(3n+2) = A041046(5n+3) and a(3n+3) = 2*A041046 (5n+4).
(End)
Sequence in context: A041571 A042259 A100105 * A097565 A079716 A203195
Adjacent sequences: A087127 A087128 A087129 * A087131 A087132 A087133
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Aug 16 2003
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