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A041010
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Numerators of continued fraction convergents to sqrt(8).
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7
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2, 3, 14, 17, 82, 99, 478, 577, 2786, 3363, 16238, 19601, 94642, 114243, 551614, 665857, 3215042, 3880899, 18738638, 22619537, 109216786, 131836323, 636562078, 768398401, 3710155682, 4478554083, 21624372014, 26102926097, 126036076402, 152139002499
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OFFSET
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0,1
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LINKS
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FORMULA
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a(n) = 6*a(n-2) - a(n-4).
a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 4*a(2n) + a(2n-1).
G.f.: (2+3*x+2*x^2-x^3)/(1-6*x^2+x^4).
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = -((3-2*sqrt(2))^n*(1+sqrt(2))) + (-1+sqrt(2))*(3+2*sqrt(2))^n.
a1(n) = ((3-2*sqrt(2))^n + (3+2*sqrt(2))^n)/2. (End)
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MATHEMATICA
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CoefficientList[Series[(2 + 3*x + 2*x^2 - x^3)/(1 - 6*x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 28 2013 *)
a0[n_] := -((3-2*Sqrt[2])^n*(1+Sqrt[2]))+(-1+Sqrt[2])*(3+2*Sqrt[2])^n // Simplify
a1[n_] := ((3-2*Sqrt[2])^n+(3+2*Sqrt[2])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &, Range[20]]] (* Gerry Martens, Jul 11 2015 *)
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PROG
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(PARI) A041010=contfracpnqn(c=contfrac(sqrt(8)), #c)[1, ][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A041010[n+1]! For more terms use:
extend(A, c, N)={for(n=#A+1, #A=Vec(A, N), A[n]=A[n-#c..n-1]*c); A} \\ (End)
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CROSSREFS
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Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A005667 (m=10), A041014 (m=11), A041016 (m=12), ..., A042934 (m=999), A042936 (m=1000).
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KEYWORD
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nonn,cofr,frac,easy
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AUTHOR
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EXTENSIONS
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Initial term 1 removed and b-file, program and formulas adapted by Georg Fischer, Jul 01 2019
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STATUS
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approved
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