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A055819
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Row sums of array T in A055818; twice the odd-indexed Fibonacci numbers after initial term.
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15
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1, 2, 4, 10, 26, 68, 178, 466, 1220, 3194, 8362, 21892, 57314, 150050, 392836, 1028458, 2692538, 7049156, 18454930, 48315634, 126491972, 331160282, 866988874, 2269806340, 5942430146, 15557484098, 40730022148, 106632582346
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OFFSET
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0,2
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COMMENTS
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Solutions (x, y) = (a(n), a(n+1)) satisfying x^2 + y^2 = 3xy - 4. - Michel Lagneau, Feb 01 2014
Except for the first term, positive values of x (or y) satisfying x^2 - 18xy + y^2 + 256 = 0. - Colin Barker, Feb 16 2014
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LINKS
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FORMULA
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a(n) = 3*a(n-1) - a(n-2) for n > 0.
G.f.: (1 -x -x^2)/(1-3*x+x^2). (End)
a(n) = 2*(ChebyshevU(n, 3/2) - 2*ChebyshevU(n-1, 3/2)), with a(0)=1.
E.g.f.: -1 + 2*exp(3*x/2)*( cosh(sqrt(5)*x/2) - sinh(sqrt(5)*x/2)/sqrt(5) ). (End)
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MAPLE
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seq(`if`(n=0, 1, simplify(2*(ChebyshevU(n, 3/2)-2*ChebyshevU(n-1, 3/2)))), n = 0..30); # G. C. Greubel, Jan 22 2020
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MATHEMATICA
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CoefficientList[Series[(1-x-x^2)/(1-3*x+x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 05 2014 *)
Join[{1}, LinearRecurrence[{3, -1}, {2, 4}, 30]] (* Harvey P. Dale, Oct 01 2014 *)
Table[If[n==0, 1, 2*(ChebyshevU[n, 3/2] -2*ChebyshevU[n-1, 3/2])], {n, 0, 30}] (* G. C. Greubel, Jan 22 2020 *)
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PROG
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(PARI) Vec((1-x-x^2)/(1-3*x+x^2) + O(x^40)) \\ Colin Barker, Feb 01 2014
(Magma) I:=[2, 4]; [1] cat [n le 2 select I[n] else 3*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 22 2020
(Sage) [1]+[2*(chebyshev_U(n, 3/2) -2*chebyshev_U(n-1, 3/2)) for n in (1..30)] # G. C. Greubel, Jan 22 2020
(GAP) a:=[2, 4];; for n in [3..30] do a[n]:=3*a[n-1]-a[n-2]; od; Concatenation([1], a); # G. C. Greubel, Jan 22 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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