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A294157 Fibonacci sequence beginning 2, 8.

%I A294157 #31 Sep 08 2022 08:46:20
%S A294157 2,8,10,18,28,46,74,120,194,314,508,822,1330,2152,3482,5634,9116,
%T A294157 14750,23866,38616,62482,101098,163580,264678,428258,692936,1121194,
%U A294157 1814130,2935324,4749454,7684778,12434232,20119010,32553242,52672252,85225494,137897746,223123240
%N A294157 Fibonacci sequence beginning 2, 8.
%D A294157 Steven Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Dover Publications (2008), page 24 (formula 8).
%H A294157 Colin Barker, <a href="/A294157/b294157.txt">Table of n, a(n) for n = 0..1000</a>
%H A294157 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A294157 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A294157 G.f.: 2*(1 + 3*x)/(1 - x - x^2).
%F A294157 a(n) = a(n-1) + a(n-2).
%F A294157 a(n) = 2*A000285(n).
%F A294157 Let g(r,s;n) be the n-th generalized Fibonacci number with initial values r, s. We have:
%F A294157 a(n) =     Lucas(n) + g(0,7;n), see A022090;
%F A294157 a(n) = Fibonacci(n) + g(2,7;n), see A022113;
%F A294157 a(n) =   2*g(1,8;n) - g(0,8;n);
%F A294157 a(n) =     g(1,k;n) + g(1,8-k;n) for all k in Z.
%F A294157 a(h+k) = a(h)*Fibonacci(k-1) + a(h+1)*Fibonacci(k) for all h, k in Z (see S. Vajda in References section). For h=0 and k=n:
%F A294157 a(n) = 2*Fibonacci(n-1) + 8*Fibonacci(n).
%F A294157 Sum_{j=0..n} a(j) = a(n+2) - 8.
%F A294157 a(n) = (2^(-n)*((1-sqrt(5))^n*(-7+sqrt(5)) + (1+sqrt(5))^n*(7+sqrt(5)))) / sqrt(5). - _Colin Barker_, Oct 25 2017
%p A294157 f:= gfun:-rectoproc({a(n)=a(n-1)+a(n-2),a(0)=2,a(1)=8},a(n),remember):
%p A294157 map(f, [$0..100]); # _Robert Israel_, Oct 24 2017
%t A294157 LinearRecurrence[{1, 1}, {2, 8}, 40]
%o A294157 (Magma) a0:=2; a1:=8; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]];
%o A294157 (PARI) Vec(2*(1 + 3*x)/(1 - x - x^2) + O(x^40)) \\ _Colin Barker_, Oct 25 2017
%o A294157 (Sage)
%o A294157 a = BinaryRecurrenceSequence(1, 1, 2, 8)
%o A294157 print([a(n) for n in range(38)]) # _Peter Luschny_, Oct 25 2017
%Y A294157 Cf. A000032, A000045, A000285.
%Y A294157 Similar sequences listed in A294116.
%K A294157 nonn,easy
%O A294157 0,1
%A A294157 _Bruno Berselli_, Oct 24 2017