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A097838
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First differences of Chebyshev polynomials S(n,51) = A097836(n) with Diophantine property.
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5
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1, 50, 2549, 129949, 6624850, 337737401, 17217982601, 877779375250, 44749530155149, 2281348258537349, 116304011655249650, 5929223246159194801, 302274081542463685201, 15410048935419488750450, 785610221624851462587749
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OFFSET
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0,2
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COMMENTS
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(7*b(n))^2 - 53*a(n)^2 = -4 with b(n)=A097837(n) give all positive solutions of this Pell equation.
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LINKS
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FORMULA
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a(n) = ((-1)^n)*S(2*n, 7*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1 - 51*x + x^2).
a(n) = S(n, 51) - S(n-1, 51) = T(2*n+1, sqrt(53)/2)/(sqrt(53)/2), with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x) and T(n, x) Chebyshev's polynomials of the first kind, A053120.
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EXAMPLE
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All positive solutions of Pell equation x^2 - 53*y^2 = -4 are (7=7*1,1), (364=7*52,50), (18557=7*2651,2549), (946043=7*135149,129949), ...
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MATHEMATICA
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LinearRecurrence[{51, -1}, {1, 50}, 20] (* G. C. Greubel, Jan 13 2019 *)
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PROG
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(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-51*x+x^2)) \\ G. C. Greubel, Jan 13 2019
(Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(1-51*x+x^2) )); // G. C. Greubel, Jan 13 2019
(Sage) ((1-x)/(1-51*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 13 2019
(GAP) a:=[1, 50];; for n in [3..20] do a[n]:=51*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 13 2019
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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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