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A136211
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Denominators in continued fraction [1, 3, 1, 3, 1, 3,...].
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4
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1, 4, 5, 19, 24, 91, 115, 436, 551, 2089, 2640, 10009, 12649, 47956, 60605, 229771, 290376, 1100899, 1391275, 5274724, 6665999, 25272721, 31938720, 121088881, 153027601, 580171684, 733199285, 2779769539, 3512968824
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| A136210(n)/A136211(n) tends to .791287847... = [1, 3, 1, 3, 1, 3,...] = (sqrt(21) - 3)/2 = the inradius of a right triangle with hypotenuse 3, legs 2 and sqrt(21).
The number .791287847... = (sqrt(21) - 3)/2 arises in finding a number which is 5 less than its square; the result is: 2.791287847... because (2.791287847...)^2 = 7.791287847... In general the quadratic equation for finding such numbers is x^2 - x = N, so x = (1 + sqrt(1 + 4N))/2. - Alexander R. Povolotsky (pevnev(AT)juno.com), Dec 23 2007
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FORMULA
| a(1) = 1, a(2) = 4, then for n>2, a(2n) = 3*a(2n-1) + a(2n-2); a(2n-1) = a(2n-2) + a(2n-3). Let T = the 2 X 2 matrix [1, 3; 1, 4]. Then T^n = [A136210(2n-1), A136210(2n); a(2n-1), a(2n)].
O.g.f.: x*(1+4*x-x^3)/(1-5*x^2+x^4). a(2n)=A004253(n+1). a(2n+1)=A004254(n). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 18 2008
a(n)*a(n+1)= A099025(n). - R. Guy (rkg(AT)cpsc.ucalgary.ca), May 18 2008
Ogf([1, 4, 5, 19, 24, 91, 115, 436, 551, 2089, 2640, 10009, 12649, 47956]) = (-x^3 + 4*x + 1)/(x^4 - 5*x^2 + 1) - Alexander R. Povolotsky (pevnev(AT)juno.com), Apr 26 2008
{-a(n) + 5 a(n + 2) - a(n + 4), a(0) = 1, a(1) = 4, a(2) = 5, a(3) = 19}. - Robert Israel, May 14 2008
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EXAMPLE
| a(4) = 19 = 3*a(3) + a(2) = 3*5 + 4.
a(5) = 24 = a(4) + a(3) = 19 + 5.
T^3 = [19, 72; 24, 91], where the bottom row [24, 91] = [a(5), a(6)].
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MATHEMATICA
| Denominator[NestList[(3/(3+#))&, 0, 60]] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 13 2010]
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CROSSREFS
| Cf. A136210.
Sequence in context: A026755 A042885 A042085 * A041036 A041699 A154704
Adjacent sequences: A136208 A136209 A136210 * A136212 A136213 A136214
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KEYWORD
| nonn,frac
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2007
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 18 2008
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