%I #111 Jun 03 2024 16:13:00
%S 1,5,34,233,1597,10946,75025,514229,3524578,24157817,165580141,
%T 1134903170,7778742049,53316291173,365435296162,2504730781961,
%U 17167680177565,117669030460994,806515533049393,5527939700884757,37889062373143906,259695496911122585,1779979416004714189
%N a(n) = Fibonacci(4*n + 1).
%C For positive n, a(n) equals (-1)^n times the permanent of the (4n) X (4n) tridiagonal matrix with sqrt(i)'s along the three central diagonals, where i is the imaginary unit. - _John M. Campbell_, Jul 12 2011
%C a(n) = 5^n*a(n; 3/5) = (16/5)^n*a(2*n; 3/4), and F(4*n) = 5^n*b(n; 3/5) = (16/5)^n*b(2*n; 3/4), where a(n; d) and b(n; d), n=0, 1, ..., d in C, denote the delta-Fibonacci numbers defined in comments to A014445. Two of these identities from the following relations follows: F(k+1)^n*a(n; F(k)/F(k+1)) = F(k*n+1) and F(k+1)^n*b(n; F(k)/F(k+1)) = F(k*n) (see also Witula's et al. papers). - _Roman Witula_, Jul 24 2012
%H Bruno Berselli, <a href="/A033889/b033889.txt">Table of n, a(n) for n = 0..500</a>
%H Edyta Hetmaniok, Bożena Piątek, and Roman Wituła, <a href="https://doi.org/10.1515/math-2017-0047">Binomials Transformation Formulae of Scaled Fibonacci Numbers</a>, Open Mathematics, 15(1) (2017), 477-485.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H Kai Wan, <a href="https://www.fq.math.ca/60-3.html">Problem H-90</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 60, No. 3 (2022), p. 282; <a href="https://www.fq.math.ca/Problems/AdvProbFeb2024.pdf">Solution to Problem H-90</a>, by Ángel Plaza, ibid., Vol. 62, No. 1 (2024), pp. 95-96.
%H Roman Wituła, <a href="https://doi.org/10.1515/dema-2013-0436">Binomials Transformation Formulae of Scaled Lucas Numbers</a>, Demonstratio Mathematica, Vol. XLVI, No. 1 (2013), pp. 15-27.
%H Roman Witula and Damian Slota, <a href="http://dx.doi.org/10.2298/AADM0902310W">delta-Fibonacci numbers</a>, Appl. Anal. Discr. Math 3 (2009), 310-329, <a href="http://www.ams.org/mathscinet-getitem?mr=2555042">MR2555042</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-1).
%F a(n) = 7*a(n-1) - a(n-2) for n >= 2. - _Floor van Lamoen_, Dec 10 2001
%F From _R. J. Mathar_, Jan 17 2008: (Start)
%F O.g.f.: (1 - 2*x)/(1 - 7*x + x^2).
%F a(n) = A004187(n+1) - 2*A004187(n). (End); corrected by _Klaus Purath_, Jul 29 2020
%F a(n) = A167816(4*n+1). - _Reinhard Zumkeller_, Nov 13 2009
%F a(n) = sqrt(1 + 2 * Fibonacci(2*n) * Fibonacci(2*n + 1) + 5 * (Fibonacci(2*n) * Fibonacci(2*n + 1))^2). - _Artur Jasinski_, Feb 06 2010
%F a(n) = Sum_{k=0..n} A122070(n,k)*2^k. - _Philippe Deléham_, Mar 13 2012
%F a(n) = Fibonacci(2*n)^2 + Fibonacci(2*n)*Fibonacci(2*n+2) + 1. - _Gary Detlefs_, Apr 18 2012
%F a(n) = Fibonacci(2*n)^2 + Fibonacci(2*n+1)^2. - _Bruno Berselli_, Apr 19 2012
%F a(n) = Sum_{k = 0..n} A238731(n,k)*4^k. - _Philippe Deléham_, Mar 05 2014
%F a(n) = A000045(A016813(n)). - _Michel Marcus_, Mar 05 2014
%F 2*a(n) = Fibonacci(4*n) + Lucas(4*n). - _Bruno Berselli_, Oct 13 2017
%F a(n) = A094567(n-1) + A094567(n), assuming A094567(-1) = 0. - _Klaus Purath_, Jul 29 2020
%F Sum_{n>=0} (-1)^n * arctan(3/a(n)) = Pi/4 (A003881) (Wan, 2022). - _Amiram Eldar_, Mar 01 2024
%F E.g.f.: exp(7*x/2)*(5*cosh(3*sqrt(5)*x/2) + sqrt(5)*sinh(3*sqrt(5)*x/2))/5. - _Stefano Spezia_, Jun 03 2024
%t Table[Fibonacci[4*n+1], {n,0,14}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2008 *)
%o (Magma) [Fibonacci(4*n+1): n in [0..100]]; // _Vincenzo Librandi_, Apr 16 2011
%o (PARI) a(n)=fibonacci(4*n+1) \\ _Charles R Greathouse IV_, Jul 15 2011
%o (PARI) Vec((1-2*x)/(1-7*x+x^2) + O(x^100)) \\ _Altug Alkan_, Dec 10 2015
%Y Cf. A000032, A000045, A004187, A014445, A016813, A122070, A167816, A094567.
%Y Cf. A003881, A049684, A081018, A081016, A081068, A172968.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_