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%I #36 May 25 2023 17:56:44
%S 2,7,51,364,2599,18557,132498,946043,6754799,48229636,344362251,
%T 2458765393,17555720002,125348805407,894997357851,6390330310364,
%U 45627309530399,325781497023157,2326097788692498,16608466017870643,118585359913786999,846705985414379636
%N a(n) = 7*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 7.
%C a(n+1)/a(n) converges to (7+sqrt(53))/2 = 7.14005... = A176439.
%C Lim a(n)/a(n+1) as n approaches infinity = 0.1400549... = 2/(7+sqrt(53)) = (sqrt(53)-7)/2 = 1/A176439 = A176439 - 7.
%C From _Johannes W. Meijer_, Jun 12 2010: (Start)
%C In general sequences with recurrence a(n) = k*a(n-1)+a(n-2) with a(0)=2 and a(1)=k [and a(-1)=0] have generating function (2-k*x)/(1-k*x-x^2). If k is odd (k>=3) they satisfy a(3n+1) = b(5n), a(3n+2)=b(5*n+3), a(3n+3)=2*b(5n+4) where b(n) is the sequence of numerators of continued fraction convergents to sqrt(k^2+4). [If k is even then a(n)/2, for n>=1, is the sequence of numerators of continued fraction convergents to sqrt(k^2/4+1).]
%C For the sequence given above k=7 which implies that it is associated with A041090.
%C For a similar statement about sequences with recurrence a(n) = k*a(n-1)+a(n-2) but with a(0)=1 [and a(-1)=0] see A054413; a sequence that is associated with A041091.
%C For more information follow the Khovanova link and see A087130, A140455 and A178765.
%C (End)
%H Vincenzo Librandi, <a href="/A086902/b086902.txt">Table of n, a(n) for n = 0..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,1).
%F a(n) = ((7+sqrt(53))/2)^n + ((7-sqrt(53))/2)^n.
%F E.g.f. : 2exp(7x/2)cosh(sqrt(53)x/2); a(n)=2^(1-n)sum{k=0..floor(n/2), C(n, 2k)53^k7^(n-2k)}. a(n)=2T(n, 7i/2)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - _Paul Barry_, Nov 15 2003
%F G.f.: (2-7x)/(1-7x-x^2). - _Philippe Deléham_, Nov 16 2008
%F From _Johannes W. Meijer_, Jun 12 2010: (Start)
%F a(2n+1) = 7*A097837(n), a(2n) = A099368(n).
%F a(3n+1) = A041090(5n), a(3n+2) = A041090(5*n+3), a(3n+3) = 2*A041090(5n+4).
%F Limit(a(n+k)/a(k), k=infinity) = (A086902(n) + A054413(n-1)*sqrt(53))/2.
%F Limit(A086902(n)/A054413(n-1), n=infinity) = sqrt(53). (End)
%e a(4) = 7*a(3) + a(2) = 7*364 + 51 = 2599.
%t RecurrenceTable[{a[0] == 2, a[1] == 7, a[n] == 7 a[n-1] + a[n-2]}, a, {n, 30}] (* _Vincenzo Librandi_, Sep 19 2016 *)
%t LinearRecurrence[{7,1},{2,7},30] (* _Harvey P. Dale_, May 25 2023 *)
%o (PARI) a(n)=([0,1; 1,7]^n*[2;7])[1,1] \\ _Charles R Greathouse IV_, Apr 06 2016
%o (Magma) I:=[2,7]; [n le 2 select I[n] else 7*Self(n-1)+Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Sep 19 2016
%Y Cf. A058316, A176439.
%Y Cf. A000032 (k=1), A006497 (k=3), A087130 (k=5), A086902 (k=7), A087798 (k=9), A001946 (k=11), A088316 (k=13), A090301 (k=15), A090306 (k=17). - _Johannes W. Meijer_, Jun 12 2010
%K nonn,easy
%O 0,1
%A Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 18 2003
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