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%I #59 Jul 18 2022 10:00:44
%S 2,7,9,16,25,41,66,107,173,280,453,733,1186,1919,3105,5024,8129,13153,
%T 21282,34435,55717,90152,145869,236021,381890,617911,999801,1617712,
%U 2617513,4235225,6852738,11087963,17940701,29028664,46969365,75998029,122967394
%N Fibonacci sequence beginning 2, 7.
%D H. S. M. Coxeter, Introduction to Geometry, Second Edition, Wiley Classics Library Edition Published 1989, p. 172.
%H Ivan Panchenko, <a href="/A022113/b022113.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/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F From _Colin Barker_, Oct 18 2013: (Start)
%F G.f.: -(5*x + 2)/(x^2 + x - 1).
%F a(n) = a(n-1) + a(n-2). (End)
%F a(n) = ((5+6*sqrt(5))/5)*((1+sqrt(5))/2)^n + ((5-6*sqrt(5))/5)*((1-sqrt(5))/2)^n starting at n=0. - _Bogart B. Strauss_, Oct 27 2013
%F a(n) = h*Fibonacci(n+k) + Fibonacci(n+k-h) with h=5, k=1. - _Bruno Berselli_, Feb 20 2017
%F a(n) = 8*F(n) + F(n-3) for F = A000045. - _J. M. Bergot_, Jul 14 2017
%F a(n) = Fibonacci(n+4) + Lucas(n-1). - _Greg Dresden_ and Henry Sauer, Mar 04 2022
%F E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 6*sqrt(5)*sinh(sqrt(5)*x/2))/5. - _Stefano Spezia_, Jul 18 2022
%t RecurrenceTable[{a[0] == 2, a[1] == 7, a[n] == a[n - 1] + a[n - 2]}, a, {n, 0, 40}] (* _Bruno Berselli_, Mar 12 2015 *)
%t LinearRecurrence[{1, 1}, {2, 7}, 37] (* or *)
%t CoefficientList[Series[-(5 x + 2)/(x^2 + x - 1), {x, 0, 36}], x] (* _Michael De Vlieger_, Jul 14 2017 *)
%o (Magma) a0:=2; a1:=7; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // _Bruno Berselli_, Feb 12 2013
%o (PARI) a(n)=8*fibonacci(n)+fibonacci(n-3) \\ _Charles R Greathouse IV_, Jul 14 2017
%o (PARI) a(n)=([0,1; 1,1]^n*[2;7])[1,1] \\ _Charles R Greathouse IV_, Jul 14 2017
%Y Cf. A000032. A000045.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
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