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Search: id:A097837
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| A097837 |
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Chebyshev polynomials S(n,51) + S(n-1,51) with Diophantine property. |
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3
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| 1, 52, 2651, 135149, 6889948, 351252199, 17906972201, 912904330052, 46540213860451, 2372638002552949, 120957997916339948, 6166485255730784399, 314369790044353664401, 16026692807006306100052, 817046963367277257438251
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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(7*a(n))^2 - 53*b(n)^2 = -4 with b(n)=A097838(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)= S(n, 51) + S(n-1, 51) = S(2*n, sqrt(53)), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x). S(n, 51)=A097836(n).
a(n)= (-2/7)*I*((-1)^n)*T(2*n+1, 7*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-51*x+x^2).
a(n)=51*a(n-1)-a(n-2) ; a(0)=1, a(1)=52. [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: A004342 A134552 A004296 this_sequence A042301 A027550 A006179
Adjacent sequences: A097834 A097835 A097836 this_sequence A097838 A097839 A097840
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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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