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%I #30 Sep 05 2019 05:09:22
%S 1,3,17,48,271,765,4319,12192,68833,194307,1097009,3096720,17483311,
%T 49353213,278635967,786554688,4440692161,12535521795,70772438609,
%U 199781794032,1127918325583,3183973182717,17975920770719,50743789129440
%N Numbers k such that 7*k^2 = floor(k*sqrt(7)*ceiling(k*sqrt(7))).
%C This is a strong divisibility sequence, that is, GCD(a(n),a(m)) = a(GCD(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - _Peter Bala_, Sep 01 2019
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lehmer_sequence">Lehmer sequence</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,16,0,-1).
%F a(1)=1, a(2)=3, a(2n) = 6*a(2n-1)-a(2n-2); a(2n+1) = 3*a(2n)-a(2n-1).
%F a(n)*a(n+3) = -3 + a(n+1)*a(n+2).
%F G.f.: x*(1+3*x+x^2)/(1-16*x^2+x^4). [corrected by _Harvey P. Dale_, Oct 31 2011]
%F a(n) = 16*a(n-2) - a(n-4), n > 4. - _Harvey P. Dale_, Oct 31 2011
%F a(n) = U_n(sqrt(18),1) = (alpha^n - beta^n)/(alpha - beta) for n odd and a(n) = 3*U_n(sqrt(18),1) = (sqrt(2)/2)*(alpha^n - beta^n)/(alpha - beta) for n even, where U_n(sqrt(R),Q) denotes the Lehmer sequence with parameters R and Q and alpha = (sqrt(3) + sqrt(14))/2 and beta = (sqrt(3) - sqrt(14))/2. - _Peter Bala_, Sep 01 2019
%t CoefficientList[Series[(1+3x+x^2)/(1-16x^2+x^4),{x,0,30}],x] (* or *) LinearRecurrence[{0,16,0,-1},{1,3,17,48},31] (* _Harvey P. Dale_, Oct 31 2011 *)
%Y Cf. A159678, A001080, A001653, A001353, A060645, A001078, A001109, A084068, A084070, A143643.
%K nonn
%O 1,2
%A _Benoit Cloitre_, May 10 2003
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