cp's OEIS Frontend

A236428 a(n) = F(n+1)^2 + F(n+1)*F(n) - F(n)^2, where F = A000045.

%I A236428 #128 May 31 2025 05:18:08
%S A236428 1,1,5,11,31,79,209,545,1429,3739,9791,25631,67105,175681,459941,
%T A236428 1204139,3152479,8253295,21607409,56568929,148099381,387729211,
%U A236428 1015088255,2657535551,6957518401,18215019649,47687540549,124847601995,326855265439,855718194319
%N A236428 a(n) = F(n+1)^2 + F(n+1)*F(n) - F(n)^2, where F = A000045.
%C A236428 (a(n) + a(n+1))/2 = F(2n+2).
%C A236428 a(n) = -a(-n-1), using the negative Fibonacci values.
%C A236428 First differences equal 2*A059929.
%C A236428 Partial sums equal A192873.
%C A236428 Unlike Fibonacci, the divisibility of a(n) by the primes is quite limited, specifically to p = 5, 11, 19, 31, 59, 71, 79, 109, ... where those after 5 are only a subset of primes congruent to {1,4} mod 5.
%C A236428 Values of a(n) mod p, for all primes p exhibit repeating pattern cycles of length k = (p-1)/m or (p+1)/m (except p = 5), based on whether p is congruent to {1,4} mod 5 or {2,3} mod 5. For p = 5, k = 2p = 10. Only the slightest similarity exists here with Fibonacci: there are formulas like this for a cycle length k, but for Fibonacci those are "divisibility cycles" for prime p, not the "pattern cycles" on mod p, and the m values differ for many primes, creating different cycle lengths for the same p.
%C A236428 a(n) has the property: a(k/2 + i) mod p + a(k/2 - 1 - i) mod p = p or 0, for all primes p, and all i 0 <= i <= k/2, in every cycle of length k. Thus, when plotted, the lower and upper halves of a every cycle have an inverted (i.e., flipped) symmetry.
%C A236428 For some primes (e.g., 13, 17, 37, 53, 61, 89, 97) each half-cycle (of length k/2) is internally symmetric (i.e., the second quarter-cycle is a mirror image of the first quarter cycle, and the fourth is a mirror image of the third, on each side of some value at k/4), while the flipped symmetry still holds for the upper and lower halves. See example for p = 61, with k = 30 in pdf file below.
%C A236428 No such symmetries on mod p, of either type, exist for Fibonacci.
%C A236428 a(n) is also (apart from sign) the determinant of a 2 X 2 matrix of squares of successive Fibonacci numbers: a(n) = (-1)^(n)*(F(n+2)^2*F(n-1)^2 -F(n)^2*F(n+1)^2). - _R. M. Welukar_, Aug 30 2014
%C A236428 For n>1 a(n) is the ceiling of the maximum area of a quadrilateral having sides of length in increasing order F(n), F(n+1), L(n), and L(n+1) with L(n)=A000032(n). - _J. M. Bergot_, Jan 19 2016
%C A236428 For n>1 a(n) is the numerator of the continued fraction [1, 1, ... 1, 2, 1, 1, ... 1, 2] with n-2 1's before each 2. - _Greg Dresden_ and _Kevin Zhanming Zheng_, Aug 16 2020
%C A236428 a(n) is the number of edge covers in the rocket graph R_{3,n+1,n}. A rocket graph R_{m,i,j} is a m-cycle with two paths attached to adjacent vertices of the cycle, which have lengths i and j respectively. This is similar to a tadpole graph but with two tails. - _Bridget Rozema_, Oct 09 2024
%H A236428 Colin Barker, <a href="/A236428/b236428.txt">Table of n, a(n) for n = 0..1000</a>
%H A236428 R. C. Alperin, <a href="https://www.fq.math.ca/Papers/57-4/alperin07132019.pdf">A family of nonlinear recurrences and their linear solutions</a>, Fib. Q., 57:4 (2019), 318-321.
%H A236428 R. C. Alperin, <a href="https://www.fq.math.ca/Papers/58-2/alperin09212019.pdf">A nonlinear recurrence and its relations to Chebyshev polynomials</a>, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
%H A236428 Richard R. Forberg, <a href="/A236428/a236428.pdf">Plot of a(n) mod 61</a>
%H A236428 Bridget Rozema and Maisie Smith, <a href="https://math.unl.edu/ncuwm/presentation-library/2024/NCUWM-2024-Poster-Rozema.pdf">Edge Covers of Unions of Path and Cycle Graphs</a>, 2024.
%H A236428 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A236428 a(n) = A001654(n) + A226205(n+1).
%F A236428 G.f.: (x^2 - x + 1)/((x + 1)*(x^2 - 3*x + 1)). - _Joerg Arndt_, Feb 23 2014
%F A236428 a(n) = (2*Lucas(2*n+1) + 3*(-1)^n)/5. - _Ralf Stephan_, Feb 27 2014
%F A236428 a(n) = 2*a(n-1) + 2*a(n-2)-a(n-3). - _Colin Barker_, Dec 20 2014
%F A236428 a(n) = F(n-1)*F(n+2) + F(n)*F(n+1). - _J. M. Bergot_, Dec 20 2014
%F A236428 a(n) = 2*F(n)*F(n+1) + (-1)^n. - _Bruno Berselli_, Oct 30 2015
%F A236428 a(n) = F(2*n+1) - F(n-1)^2 +(-1)^n for n>0. - _J. M. Bergot_, Jan 19 2016
%F A236428 a(n) = (2^(-n)*(3*(-2)^n-(3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n))/5. - _Colin Barker_, Sep 28 2016
%F A236428 a(n) = F(n+1)^2 + F(n)*F(n-1). See also A099016, tenth formula. - _Bruno Berselli_, Feb 15 2017
%F A236428 2*a(n) = L(n)*L(n+1) - F(n)*F(n+1), where L = A000032. - _Bruno Berselli_, Sep 27 2017
%t A236428 a[n_] := Fibonacci[n+1]^2 + Fibonacci[n+1]*Fibonacci[n] - Fibonacci[n]^2; Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Feb 27 2014 *)
%t A236428 LinearRecurrence[{2, 2, -1}, {1, 1, 5}, 40] (* _Vincenzo Librandi_, Jan 20 2016 *)
%o A236428 (PARI) F=fibonacci;
%o A236428 a(n)=F(n+1)^2 + F(n+1)*F(n) - F(n)^2;
%o A236428 vector(33,n,a(n-1)) \\ _Joerg Arndt_, Feb 23 2014
%o A236428 (PARI) Vec((x^2-x+1)/((x+1)*(x^2-3*x+1)) + O(x^100)) \\ _Colin Barker_, Dec 20 2014
%o A236428 (PARI) a(n) = round((2^(-n)*(3*(-2)^n-(3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n))/5) \\ _Colin Barker_, Sep 28 2016
%o A236428 (Magma) [Fibonacci(n+1)^2+Fibonacci(n+1)*Fibonacci(n)- Fibonacci(n)^2: n in [0..30]]; // _Vincenzo Librandi_, Jan 20 2016
%o A236428 (Magma) F:=Fibonacci; [F(n+1)^2+F(n)*F(n-1): n in [0..30]]; // _Bruno Berselli_, Feb 15 2017
%Y A236428 Cf. A000032, A000045, A059929, A192873.
%Y A236428 Cf. similar sequences of the type k*F(n)*F(n+1)+(-1)^n listed in A264080.
%K A236428 nonn,easy
%O A236428 0,3
%A A236428 _Richard R. Forberg_, Jan 25 2014