OFFSET
0,1
COMMENTS
a(n+1)/a(n) converges to (7+sqrt(53))/2 = 7.14005... = A176439.
Lim a(n)/a(n+1) as n approaches infinity = 0.1400549... = 2/(7+sqrt(53)) = (sqrt(53)-7)/2 = 1/A176439 = A176439 - 7.
From Johannes W. Meijer, Jun 12 2010: (Start)
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).]
For the sequence given above k=7 which implies that it is associated with A041090.
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.
(End)
LINKS
FORMULA
a(n) = ((7+sqrt(53))/2)^n + ((7-sqrt(53))/2)^n.
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
G.f.: (2-7x)/(1-7x-x^2). - Philippe Deléham, Nov 16 2008
From Johannes W. Meijer, Jun 12 2010: (Start)
EXAMPLE
a(4) = 7*a(3) + a(2) = 7*364 + 51 = 2599.
MATHEMATICA
RecurrenceTable[{a[0] == 2, a[1] == 7, a[n] == 7 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)
LinearRecurrence[{7, 1}, {2, 7}, 30] (* Harvey P. Dale, May 25 2023 *)
PROG
(PARI) a(n)=([0, 1; 1, 7]^n*[2; 7])[1, 1] \\ Charles R Greathouse IV, Apr 06 2016
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 18 2003
STATUS
approved