Search: id:A077416
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%I A077416
%S A077416 1,13,155,1847,22009,262261,3125123,37239215,443745457,5287706269,
%T A077416 63008729771,750817050983,8946795882025,106610733533317,
%U A077416 1270382006517779,15137973344680031,180385298129642593
%N A077416 Chebyshev S-sequence with Diophantine property.
%C A077416 7*b(n)^2 - 5*a(n)^2 = 2 with companion sequence b(n)=A077417(n), n>=0.
%C A077416 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
%H A077416 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A077416 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A077416 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A077416 a(n) = 12*a(n-1) - a(n-2), a(-1)=-1, a(0)=1.
%F A077416 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).
%F A077416 G.f.: (1+x)/(1-12*x+x^2).
%F A077416 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).
%F A077416 a(n)= sum(((-1)^k)*binomial(2*n-k, k)*14^(n-k), k=0..n).
%F A077416 a(n) = sqrt((7*A077417(n)^2 - 2)/5).
%o A077416 (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]
%Y A077416 Cf. A054320(n-1) with companion A072256(n), n>=1.
%Y A077416 Sequence in context: A097168 A108366 A163415 this_sequence A102146 A097827 
               A142104
%Y A077416 Adjacent sequences: A077413 A077414 A077415 this_sequence A077417 A077418 
               A077419
%K A077416 nonn,easy
%O A077416 0,2
%A A077416 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 
               2002

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