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A084070
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a(n) = 38*a(n-1) - a(n-2), with a(0)=0, a(1)=6.
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6
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0, 6, 228, 8658, 328776, 12484830, 474094764, 18003116202, 683644320912, 25960481078454, 985814636660340, 37434995712014466, 1421544022419889368, 53981237856243781518, 2049865494514843808316, 77840907553707820934490, 2955904621546382351702304
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OFFSET
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0,2
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COMMENTS
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This sequence gives the values of y in solutions of the Diophantine equation x^2 - 10*y^2 = 1. The corresponding x values are in A078986. - Vincenzo Librandi, Aug 08 2010 [edited by Jon E. Schoenfield, May 04 2014]
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LINKS
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FORMULA
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Numbers k such that 10*k^2 = floor(k*sqrt(10)*ceiling(k*sqrt(10))).
a(n) = 37*(a(n-1) + a(n-2)) - a(n-3).
a(n) = 39*(a(n-1) - a(n-2)) + a(n-3). (End)
O.g.f.: 6*x/(1 - 38*x + x^2).
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EXAMPLE
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G.f. = 6*x + 228*x^2 + 8658*x^3 + 328776*x^4 + ... - Michael Somos, Feb 24 2023
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MAPLE
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seq( simplify(6*ChebyshevU(n-1, 19)), n=0..20); # G. C. Greubel, Jan 12 2020
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MATHEMATICA
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LinearRecurrence[{38, -1}, {0, 6}, 30] (* Harvey P. Dale, Nov 01 2011 *)
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PROG
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(PARI) u=0; v=6; for(n=2, 20, w=38*v-u; u=v; v=w; print1(w, ", "))
(PARI) vector(21, n, 6*polchebyshev(n-2, 2, 19) ) \\ G. C. Greubel, Jan 12 2020
(Magma) I:=[0, 6]; [n le 2 select I[n] else 38*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Jan 12 2020
(Sage) [6*chebyshev_U(n-1, 19) for n in (0..20)] # G. C. Greubel, Jan 12 2020
(GAP) a:=[0, 6];; for n in [3..20] do a[n]:=38*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 12 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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