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A136210
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Numerators in continued fraction [1, 3, 1, 3, 1, 3,...].
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2
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1, 3, 4, 15, 19, 72, 91, 345, 436, 1653, 2089, 7920, 10009, 37947, 47956, 181815, 229771, 871128, 1100899, 4173825, 5274724, 19997997, 25272721, 95816160, 121088881, 459082803, 580171684, 2199597855, 2779769539, 10538906472
(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 .7912878474... = (sqrt(21) - 3)/2 = continued fraction [1, 3, 1, 3, 1, 3,...] = the inradius of a right triangle with hypotenuse 5, legs 2 and sqrt(21).
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FORMULA
| a(1) = 1, a(2n) = 3*a(2n-1) + a(2n-2); a(2n-1) = a(2n-2) + a(2n-3). Given the 2 X 2 matrix [1, 3; 1, 4] = T, [a(2n-1), a(2n)] = top row of T^n.
Empirical G.f.: x*(1+3*x-x^2)/(1-5*x^2+x^4). [Colin Barker, Jan 04 2012]
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EXAMPLE
| a(4) = 15 = 3*a(3) + a(2) = 3*4 + 3.
a(5) = 19 = a(4) + a(3) = 15 + 4.
T^3 = [19, 72; 24, 91], where [19, 72] = [a(5), a(6)]. [24, 91] = [A136211(5), A136211(6)].
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MATHEMATICA
| a = {1, 3}; Do[If[EvenQ[n], AppendTo[a, 3*a[[ -1]] + a[[ -2]]], AppendTo[a, a[[ -1]] + a[[ -2]]]], {n, 3, 30}]; a - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Dec 31 2007
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CROSSREFS
| Cf. A136211.
Sequence in context: A053359 A056742 A041435 * A041819 A095799 A109926
Adjacent sequences: A136207 A136208 A136209 * A136211 A136212 A136213
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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 Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Dec 31 2007
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