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A077416
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Chebyshev S-sequence with Diophantine property.
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10
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1, 13, 155, 1847, 22009, 262261, 3125123, 37239215, 443745457, 5287706269, 63008729771, 750817050983, 8946795882025, 106610733533317, 1270382006517779, 15137973344680031, 180385298129642593
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OFFSET
| 0,2
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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 (reinhard.zumkeller(AT)gmail.com), Jun 01 2005
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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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).
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PROG
| (Other) sage: [(lucas_number2(n, 12, 1)-lucas_number2(n-1, 12, 1))/10 for n in xrange(1, 18)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 10 2009]
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CROSSREFS
| Cf. A054320(n-1) with companion A072256(n), n>=1.
Sequence in context: A097168 A108366 A163415 * A192092 A102146 A162768
Adjacent sequences: A077413 A077414 A077415 * A077417 A077418 A077419
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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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