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Search: id:A097838
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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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4
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| 1, 50, 2549, 129949, 6624850, 337737401, 17217982601, 877779375250, 44749530155149, 2281348258537349, 116304011655249650, 5929223246159194801, 302274081542463685201, 15410048935419488750450, 785610221624851462587749
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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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Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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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.
a(n)=51*a(n-1)-a(n-2) ; a(0)=1, a(1)=50. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 18 2008]
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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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CROSSREFS
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Sequence in context: A120998 A165800 A042201 this_sequence A047683 A010564 A115436
Adjacent sequences: A097835 A097836 A097837 this_sequence A097839 A097840 A097841
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 10 2004
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