%I #52 Jan 04 2024 09:12:21

%S 0,1,2,11,36,149,550,2143,8136,31273,119498,457907,1752300,6709949,

%T 25685998,98341639,376485264,1441362001,5518120850,21125775707,

%U 80878397364,309637224677,1185423230902,4538307034543,17374576685400

%N a(n) = 2*a(n-1) + 7*a(n-2), with a(0) = 0, a(1) = 1.

%C The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 8 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(8). - _Cino Hilliard_, Sep 25 2005

%C Pisano period lengths: 1, 2, 8, 4, 24, 8, 3, 8, 24, 24, 15, 8, 168, 6, 24, 16, 16, 24, 120, 24, ... . - _R. J. Mathar_, Aug 10 2012

%D John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

%H Vincenzo Librandi, <a href="/A015519/b015519.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,7).

%F From Mario Catalani (mario.catalani(AT)unito.it), Apr 23 2003: (Start)

%F a(n) = a(n-1) + A083100(n-2), n>1.

%F A083100(n)/a(n+1) converges to sqrt(8). (End)

%F From _Paul Barry_, Jul 17 2003: (Start)

%F G.f.: x/ ( 1-2*x-7*x^2 ).

%F a(n) = ((1+2*sqrt(2))^n-(1-2*sqrt(2))^n)*sqrt(2)/8. (End)

%F E.g.f.: exp(x)*sinh(2*sqrt(2)*x)/(2*sqrt(2)). - _Paul Barry_, Nov 20 2003

%F Second binomial transform is A000129(2n)/2 (A001109). - _Paul Barry_, Apr 21 2004

%F a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k)*(7/2)^k*2^(n-k-1). - _Paul Barry_, Jul 17 2004

%F a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*8^k. - _Paul Barry_, Sep 29 2004

%F G.f.: G(0)*x/(2*(1-x)), where G(k)= 1 + 1/(1 - x*(8*k-1)/(x*(8*k+7) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 26 2013

%t LinearRecurrence[{2,7},{0,1},30] (* _Harvey P. Dale_, Oct 09 2017 *)

%o (Sage) [lucas_number1(n,2,-7) for n in range(0, 25)] # _Zerinvary Lajos_, Apr 22 2009

%o (Magma) [ n eq 1 select 0 else n eq 2 select 1 else 2*Self(n-1)+7*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Aug 23 2011

%o (PARI) a(n)=([0,1; 7,2]^n*[0;1])[1,1] \\ _Charles R Greathouse IV_, May 10 2016

%Y The following sequences (and others) belong to the same family: A000129, A001333, A002532, A002533, A002605, A015518, A015519, A026150, A046717, A063727, A083098, A083099, A083100, A084057.

%K nonn,easy

%O 0,3

%A _Olivier Gérard_