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A077416 Chebyshev S-sequence with Diophantine property. 11
1, 13, 155, 1847, 22009, 262261, 3125123, 37239215, 443745457, 5287706269, 63008729771, 750817050983, 8946795882025, 106610733533317, 1270382006517779, 15137973344680031, 180385298129642593 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

7*b(n)^2 - 5*a(n)^2 = 2 with companion sequence b(n)=A077417(n), n>=0.

a(n) = L(n,-12)*(-1)^n, where L is defined as in A108299; see also A077417 for L(n,+12). - Reinhard Zumkeller, Jun 01 2005

LINKS

Ivan Panchenko, Table of n, a(n) for n = 0..200

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (12,-1).

FORMULA

a(n) = 12*a(n-1) - a(n-2), a(-1)=-1, a(0)=1.

a(n) = S(n, 12) + S(n-1, 12) = S(2*n, sqrt(14)) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0, S(n, 12) = A004191(n).

G.f.: (1+x)/(1-12*x+x^2).

a(n) = (ap^(2*n+1) - am^(2*n+1))/(ap - am) with ap := (sqrt(7)+sqrt(5))/sqrt(2) and am := (sqrt(7)-sqrt(5))/sqrt(2).

a(n) = sum(((-1)^k)*binomial(2*n-k, k)*14^(n-k), k=0..n).

a(n) = sqrt((7*A077417(n)^2 - 2)/5).

MATHEMATICA

LinearRecurrence[{12, -1}, {1, 13}, 30] (* Harvey P. Dale, Apr 03 2013 *)

PROG

(Sage) [(lucas_number2(n, 12, 1)-lucas_number2(n-1, 12, 1))/10 for n in xrange(1, 18)] # Zerinvary Lajos, Nov 10 2009

CROSSREFS

Cf. A054320(n-1) with companion A072256(n), n>=1.

Sequence in context: A108366 A204766 A163415 * A192092 A102146 A162768

Adjacent sequences:  A077413 A077414 A077415 * A077417 A077418 A077419

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 29 2002

STATUS

approved

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Last modified November 19 08:51 EST 2017. Contains 294923 sequences.