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A088317
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a(n) = 8a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.
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2
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1, 4, 33, 268, 2177, 17684, 143649, 1166876, 9478657, 76996132, 625447713, 5080577836, 41270070401, 335241141044, 2723199198753, 22120834731068, 179689877047297, 1459639851109444
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Harmonious sequence, build on the number 8.1231056... a(n+1)/a(n) converges to 4+sqrt(17). a(0)/a(1)=1/4; a(1)/a(2)=4/33; a(2)/a(3)=33/268; a(3)/a(4)=268/2177;...etc. Lim a(n)/a(n+1)as n approaches infinity=0.123105625...=1/(4+sqrt(17))=sqrt(17)-4.
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n) = 8a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4. a(n) = ([(8+sqrt(68))/2]^n + [(8-sqrt(68))/2]^n)/2. a(n) = A086594(n)/2
E.g.f. : exp(4x)cosh(sqrt(17)x); a(n)=((4+sqrt(17))^n+(4-sqrt(17))^n)/2; a(n)=sum{k=0..floor(n/2), C(n, 2k)17^k4^(n-2k)}. a(n)=T(n, 4i)(-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
a(n)=A041024(n-1), n>0. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 11 2008]
G.f.: (1-4x)/(1-8x-x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 16 2008]
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EXAMPLE
| a(4) = 2177 = 8a(3) + a(2) = 8*268 + 33 = ([(8+sqrt(68))/2]^4 + [(8-sqrt(68))/2]^4)/2 = (4353.9997703+ 0.0002297)/2=2177.
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CROSSREFS
| Cf. A002018, A002190, A013192, A028576, A041027, A058153, A058155, A072754, A075132.
Cf. A041024.
Sequence in context: A131509 A081007 A203212 * A041024 A202765 A123780
Adjacent sequences: A088314 A088315 A088316 * A088318 A088319 A088320
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KEYWORD
| easy,nonn
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AUTHOR
| Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Nov 06 2003
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EXTENSIONS
| Corrected generating function. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 20 2008
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