cp's OEIS Frontend

A294116 Fibonacci sequence beginning 2, 21.

%I A294116 #41 Sep 08 2022 08:46:20
%S A294116 2,21,23,44,67,111,178,289,467,756,1223,1979,3202,5181,8383,13564,
%T A294116 21947,35511,57458,92969,150427,243396,393823,637219,1031042,1668261,
%U A294116 2699303,4367564,7066867,11434431,18501298,29935729,48437027,78372756,126809783,205182539,331992322,537174861
%N A294116 Fibonacci sequence beginning 2, 21.
%D A294116 Steven Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Dover Publications (2008), page 24 (formula 8).
%H A294116 Colin Barker, <a href="/A294116/b294116.txt">Table of n, a(n) for n = 0..1000</a>
%H A294116 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A294116 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A294116 G.f.: (2 + 19*x)/(1 - x - x^2).
%F A294116 a(n) = a(n-1) + a(n-2).
%F A294116 Let g(r,s;n) be the n-th generalized Fibonacci number with initial values r, s. We have:
%F A294116 a(n) =     Lucas(n) + g(0,20;n), see A022354;
%F A294116 a(n) = Fibonacci(n) + g(2,20;n), see A022372;
%F A294116 a(n) =  2*g(1,21;n) - g(0,21;n);
%F A294116 a(n) =     g(1,k;n) + g(1,21-k;n) for all k in Z.
%F A294116 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 A294116 a(n) = 2*Fibonacci(n-1) + 21*Fibonacci(n).
%F A294116 Sum_{j=0..n} a(j) = a(n+2) - 21.
%F A294116 a(n) = (2^(-n)*((1-sqrt(5))^n*(-20+sqrt(5)) + (1+sqrt(5))^n*(20+sqrt(5)))) / sqrt(5). - _Colin Barker_, Oct 25 2017
%t A294116 LinearRecurrence[{1, 1}, {2, 21}, 40]
%o A294116 (Magma) a0:=2; a1:=21; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]];
%o A294116 (PARI) Vec((2 + 19*x)/(1 - x - x^2) + O(x^40)) \\ _Colin Barker_, Oct 25 2017
%o A294116 (Sage)
%o A294116 a = BinaryRecurrenceSequence(1, 1, 2, 21)
%o A294116 print([a(n) for n in range(38)]) # _Peter Luschny_, Oct 25 2017
%Y A294116 Cf. A000032, A000045.
%Y A294116 Subsequence of A047201, A047592, A113763.
%Y A294116 Sequences of the type g(2,k;n): A118658 (k=0), A000032 (k=1), 2*A000045 (k=2,4), A020695 (k=3), A001060 (k=5), A022112 (k=6), A022113 (k=7), A294157 (k=8), A022114 (k=9), A022367 (k=10), A022115 (k=11), A022368 (k=12), A022116 (k=13), A022369 (k=14), A022117 (k=15), A022370 (k=16), A022118 (k=17), A022371 (k=18), A022119 (k=19), A022372 (k=20), this sequence (k=21), A022373 (k=22); A022374 (k=24); A022375 (k=26); A022376 (k=28), A190994 (k=29), A022377 (k=30); A022378 (k=32).
%K A294116 nonn,easy
%O A294116 0,1
%A A294116 _Bruno Berselli_, Oct 23 2017