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A033890 a(n) = Fibonacci(4*n + 2).

%I A033890 #145 Aug 24 2025 21:52:50
%S A033890 1,8,55,377,2584,17711,121393,832040,5702887,39088169,267914296,
%T A033890 1836311903,12586269025,86267571272,591286729879,4052739537881,
%U A033890 27777890035288,190392490709135,1304969544928657,8944394323791464,61305790721611591,420196140727489673
%N A033890 a(n) = Fibonacci(4*n + 2).
%C A033890 (x,y) = (a(n), a(n+1)) are solutions of (x+y)^2/(1+xy)=9, the other solutions are in A033888. - _Floor van Lamoen_, Dec 10 2001
%C A033890 This sequence consists of the odd-indexed terms of A001906 (whose terms are the values of x such that 5*x^2 + 4 is a square). The even-indexed terms of A001906 are in A033888. Limit_{n->infinity} a(n)/a(n-1) = phi^4 = (7 + 3*sqrt(5))/2. - _Gregory V. Richardson_, Oct 13 2002
%C A033890 General recurrence is a(n) = (a(1)-1)*a(n-1) - a(n-2), a(1) >= 4, lim_{n->infinity} a(n) = x*(k*x+1)^n, k = a(1) - 3, x = (1 + sqrt((a(1)+1)/(a(1)-3)))/2. Examples in OEIS: a(1)=4 gives A002878. a(1)=5 gives A001834. a(1)=6 gives A030221. a(1)=7 gives A002315. a(1)=8 gives A033890. a(1)=9 gives A057080. a(1)=10 gives A057081. - _Ctibor O. Zizka_, Sep 02 2008
%C A033890 Indices of square numbers which are also 12-gonal. - _Sture Sjöstedt_, Jun 01 2009
%C A033890 For positive n, a(n) equals the permanent of the (2n) X (2n) tridiagonal matrix with 3's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - _John M. Campbell_, Jul 08 2011
%C A033890 If we let b(0) = 0 and, for n >= 1, b(n) = A033890(n-1), then the sequence b(n) will be F(4n-2) and the first difference is L(4n) or A056854. F(4n-2) is also the ratio of golden spiral length (rounded to the nearest integer) after n rotations. L(4n) is also the pitch length ratio. See illustration in links. - _Kival Ngaokrajang_, Nov 03 2013
%C A033890 The aerated sequence (b(n))n>=1 = [1, 0, 8, 0, 55, 0, 377, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -5, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047. - _Peter Bala_, Mar 22 2015
%C A033890 Solutions y of Pell equation x^2 - 5*y^2 = 4; corresponding x values are in A342710 (see A342709). - _Bernard Schott_, Mar 19 2021
%H A033890 G. C. Greubel, <a href="/A033890/b033890.txt">Table of n, a(n) for n = 0..1000</a>
%H A033890 Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.
%H A033890 Nathan D. Cahill and Darren A. Narayan, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-3/quartcahill03_2004.pdf">Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants</a>, Fib. Quart. 42, no. 3, Aug. 2004, pp. 216-221. See p. 219.
%H A033890 Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
%H A033890 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A033890 Donatella Merlini and Renzo Sprugnoli, <a href="https://doi.org/10.1016/j.disc.2016.08.017">Arithmetic into geometric progressions through Riordan arrays</a>, Discrete Mathematics 340.2 (2017): 160-174.
%H A033890 Kival Ngaokrajang, <a href="/A033890/a033890.pdf">Illustration of golden spiral length and pitch ratio</a>
%H A033890 H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H A033890 H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a>, Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
%H A033890 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-1).
%H A033890 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A033890 G.f.: (1+x)/(1-7*x+x^2).
%F A033890 a(n) = 7*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=8.
%F A033890 a(n) = S(n,7) + S(n-1,7) = S(2*n,sqrt(9) = 3), where S(n,x) = U(n,x/2) are Chebyshev's polynomials of the 2nd kind. Cf. A049310. S(n,7) = A004187(n+1), S(n,3) = A001906(n+1).
%F A033890 a(n) = ((7+3*sqrt(5))^n - (7-3*sqrt(5))^n + 2*((7+3*sqrt(5))^(n-1) - ((7-3*sqrt(5))^(n-1)))) / (3*(2^n)*sqrt(5)). - _Gregory V. Richardson_, Oct 13 2002
%F A033890 Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i); then a(n) = (-1)^n*q(n, -9). - _Benoit Cloitre_, Nov 10 2002
%F A033890 a(n) = L(n,-7)*(-1)^n, where L is defined as in A108299; see also A049685 for L(n,+7). - _Reinhard Zumkeller_, Jun 01 2005
%F A033890 Define f(x,s) = s*x + sqrt((s^2-1)*x^2+1); f(0,s)=0. a(n) = f(a(n-1),7/2) + f(a(n-2),7/2). - _Marcos Carreira_, Dec 27 2006
%F A033890 a(n+1) = 8*a(n) - 8*a(n-1) + a(n-2); a(1)=1, a(2)=8, a(3)=55. - _Sture Sjöstedt_, May 27 2009
%F A033890 a(n) = A167816(4*n+2). - _Reinhard Zumkeller_, Nov 13 2009
%F A033890 a(n)=b such that (-1)^n*Integral_{0..Pi/2} (cos((2*n+1)*x))/(3/2-sin(x)) dx = c + b*log(3). - _Francesco Daddi_, Aug 01 2011
%F A033890 a(n) = A000045(A016825(n)). - _Michel Marcus_, Mar 22 2015
%F A033890 a(n) = A001906(2*n+1). - _R. J. Mathar_, Apr 30 2017
%F A033890 E.g.f.: exp(7*x/2)*(5*cosh(3*sqrt(5)*x/2) + 3*sqrt(5)*sinh(3*sqrt(5)*x/2))/5. - _Stefano Spezia_, Apr 14 2025
%F A033890 From _Peter Bala_, Jun 08 2025: (Start)
%F A033890 Sum_{n >= 1} (-1)^(n+1)/(a(n) - 1/a(n)) = 1/9 [telescoping series: 3/(a(n) - 1/a(n)) = 1/Fibonacci(4*n+4) + 1/Fibonacci(4*n)].
%F A033890 Product_{n >= 1} (a(n) + 3)/(a(n) - 3) = 5/2 [telescoping product:
%F A033890 (a(n) + 3)/(a(n) - 3) = b(n)/b(n-1), where b(n) = (Lucas(4*n+4) - 3)/(Lucas(4*n+4) + 3)].
%F A033890 Product_{n >= 1} (a(n) + 1)/(a(n) - 1) = sqrt(9/5) [telescoping product:
%F A033890 (a(n) + 1)/(a(n) - 1) = c(n)/c(n-1) for n >= 1, where c(n) = Fibonacci(2*n+2)/Lucas(2*n+2)]. (End)
%F A033890 From _Gerry Martens_, Aug 20 2025: (Start)
%F A033890 a(n) = ((3 + sqrt(5))^(1 + 2*n) - (3 - sqrt(5))^(1 + 2*n)) / (2^(1 + 2*n)*sqrt(5)).
%F A033890 a(n) = Sum_{k=0..2*n} binomial(2*n + k + 1, 2*k + 1). (End)
%p A033890 A033890 := proc(n)
%p A033890     option remember;
%p A033890     if n <= 1 then
%p A033890         op(n+1,[1,8]);
%p A033890     else
%p A033890         7*procname(n-1)-procname(n-2) ;
%p A033890     end if;
%p A033890 end proc: # _R. J. Mathar_, Apr 30 2017
%t A033890 Table[Fibonacci[4n + 2], {n, 0, 14}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2008 *)
%t A033890 LinearRecurrence[{7, -1}, {1, 8}, 50] (* _G. C. Greubel_, Jul 13 2017 *)
%t A033890 a[n_] := (GoldenRatio^(2 (1 + 2 n)) - GoldenRatio^(-2 (1 + 2 n)))/Sqrt[5]
%t A033890 Table[a[n] // FullSimplify, {n, 0, 21}] (* _Gerry Martens_, Aug 20 2025 *)
%o A033890 (PARI) a(n)=fibonacci(4*n+2);
%o A033890 (Magma) [Fibonacci(4*n +2): n in [0..100]]; // _Vincenzo Librandi_, Apr 17 2011
%Y A033890 Cf. A000045, A001906, A100047.
%Y A033890 Cf. A342709, A342710.
%K A033890 nonn,easy,changed
%O A033890 0,2
%A A033890 _N. J. A. Sloane_