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A077420
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Bisection of Chebyshev sequence T(n,3) (odd part) with Diophantine property.
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7
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1, 33, 1121, 38081, 1293633, 43945441, 1492851361, 50713000833, 1722749176961, 58522759015841, 1988051057361633, 67535213191279681, 2294209197446147521, 77935577499977736033, 2647515425801796877601
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| (3*a(n))^2 - 2*(2*b(n))^2 = 1 with companion sequence b(n)= A046176(n+1), n>=0 (special solutions of Pell equation).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index to sequences with linear recurrences with constant coefficients, signature (34,-1).
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FORMULA
| a(n) = 34*a(n-1) - a(n-2), a(-1)=1, a(0)=1.
a(n) = T(2*n+1, 3)/3 = S(n, 34) - S(n-1, 34) with S(n, x) := U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120. S(-1, x)=0, S(n, 34)= A029547(n), T(n, 3)=A001541(n).
G.f.: (1-x)/(1-34*x+x^2).
a(n)= sqrt(8*A046176(n+1)^2 + 1)/3.
a(n) = (k^n)+(k^(-n))-a(n-1) = A003499(2n)-a(n-1)), where k = (sqrt(2)+1)^4 = 17+12*sqrt(2) and a(0)=1. - Charles L. Hohn (ch+oeis(AT)1111-internet.com), Apr 05 2011
a(n) = a(-n-1) = A029547(n)-A029547(n-1) = ((1+sqrt(2))^(4n+2)+(1-sqrt(2))^(4n+2))/6. - Bruno Berselli, Nov 22 2011
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MATHEMATICA
| LinearRecurrence[{34, -1}, {1, 33}, 20] (* Vincenzo Librandi, Nov 22 2011 *)
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PROG
| (MAGMA) I:=[1, 33]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011
(PARI) Vec((1-x)/(1-34*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Nov 22 2011
(Maxima) makelist(expand(((1+sqrt(2))^(4*n+2)+(1-sqrt(2))^(4*n+2))/6), n, 0, 14); [Bruno Berselli, Nov 22 2011]
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CROSSREFS
| Cf. A056771 (even part).
Row 34 of array A094954.
Row 3 of array A188646.
Sequence in context: A009977 A187539 A130835 * A158688 A065424 A071268
Adjacent sequences: A077417 A077418 A077419 * A077421 A077422 A077423
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002
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