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A190995 Fibonacci sequence beginning 9, 7.

%I A190995 #50 Mar 17 2025 15:49:42
%S A190995 9,7,16,23,39,62,101,163,264,427,691,1118,1809,2927,4736,7663,12399,
%T A190995 20062,32461,52523,84984,137507,222491,359998,582489,942487,1524976,
%U A190995 2467463,3992439,6459902,10452341,16912243,27364584,44276827,71641411,115918238,187559649
%N A190995 Fibonacci sequence beginning 9, 7.
%C A190995 From _Wajdi Maaloul_, Jun 20 2022: (Start)
%C A190995 For n>0, 2*a(n) is the number of ways to tile this figure below with squares and dominoes (a strip of length n+1 that begins with a length 3 vertical strip and length 4 one).
%C A190995      _
%C A190995    _|_|
%C A190995   |_|_|
%C A190995   |_|_|_______     _
%C A190995   |_|_|_|_|_|_|...|_|
%C A190995 (End)
%H A190995 Vincenzo Librandi, <a href="/A190995/b190995.txt">Table of n, a(n) for n = 0..1000</a>
%H A190995 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A190995 a(n) = ((9+sqrt(5))/2)*((1+sqrt(5))/2)^n + ((9-sqrt(5))/2)*((1-sqrt(5))/2)^n. - _Antonio Alberto Olivares_
%F A190995 G.f.: (9-2*x)/(1-x-x^2). - _Colin Barker_, Jan 11 2012
%F A190995 a(n) = 7*Fibonacci(n) + 9*Fibonacci(n-1) = 7*Fibonacci(n+1) + 2*Fibonacci(n-1) = 7*Lucas(n) - 5*Fibonacci(n-1) for n>0. - _Wajdi Maaloul_, Jun 20 2022
%p A190995 a:= n-> (<<0|1>, <1|1>>^n. <<9, 7>>)[1, 1]:
%p A190995 seq(a(n), n=0..36);  # _Alois P. Heinz_, Oct 26 2022
%t A190995 LinearRecurrence[{1, 1}, {9, 7}, 100]
%o A190995 (PARI) a(n)=7*fibonacci(n)+9*fibonacci(n-1) \\ _Charles R Greathouse IV_, Jun 08 2011
%o A190995 (Magma) [n le 2 select 11-2*n else Self(n-1)+Self(n-2): n in [1..50]]; // _Vincenzo Librandi_, Feb 15 2012
%o A190995 (SageMath) [7*fibonacci(n) + 9*fibonacci(n-1) for n in range(51)] # _G. C. Greubel_, Oct 26 2022
%Y A190995 Cf. A000032, A000045, A190994.
%K A190995 nonn,easy
%O A190995 0,1
%A A190995 _Vladimir Joseph Stephan Orlovsky_, Jun 07 2011