A099016 -id:A099016 - cp's OEIS frontend

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

1, 1, 5, 11, 31, 79, 209, 545, 1429, 3739, 9791, 25631, 67105, 175681, 459941, 1204139, 3152479, 8253295, 21607409, 56568929, 148099381, 387729211, 1015088255, 2657535551, 6957518401, 18215019649, 47687540549, 124847601995, 326855265439, 855718194319

Offset: 0

Richard R. Forberg, Jan 25 2014

(a(n) + a(n+1))/2 = F(2n+2).
a(n) = -a(-n-1), using the negative Fibonacci values.
First differences equal 2*A059929.
Partial sums equal A192873.
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.
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.
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.
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.
No such symmetries on mod p, of either type, exist for Fibonacci.
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
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
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
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

Colin Barker, Table of n, a(n) for n = 0..1000
R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.
R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
Richard R. Forberg, Plot of a(n) mod 61
Bridget Rozema and Maisie Smith, Edge Covers of Unions of Path and Cycle Graphs, 2024.
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000032, A000045, A059929, A192873.

Cf. similar sequences of the type k*F(n)*F(n+1)+(-1)^n listed in A264080.

[Fibonacci(n+1)^2+Fibonacci(n+1)*Fibonacci(n)- Fibonacci(n)^2: n in [0..30]]; // Vincenzo Librandi, Jan 20 2016

F:=Fibonacci; [F(n+1)^2+F(n)*F(n-1): n in [0..30]]; // Bruno Berselli, Feb 15 2017

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 *)
LinearRecurrence[{2, 2, -1}, {1, 1, 5}, 40] (* Vincenzo Librandi, Jan 20 2016 *)

F=fibonacci;
a(n)=F(n+1)^2 + F(n+1)*F(n) - F(n)^2;
vector(33,n,a(n-1)) \\ Joerg Arndt, Feb 23 2014

Vec((x^2-x+1)/((x+1)*(x^2-3*x+1)) + O(x^100)) \\ Colin Barker, Dec 20 2014

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

a(n) = A001654(n) + A226205(n+1).
G.f.: (x^2 - x + 1)/((x + 1)*(x^2 - 3*x + 1)). - Joerg Arndt, Feb 23 2014
a(n) = (2*Lucas(2*n+1) + 3*(-1)^n)/5. - Ralf Stephan, Feb 27 2014
a(n) = 2*a(n-1) + 2*a(n-2)-a(n-3). - Colin Barker, Dec 20 2014
a(n) = F(n-1)*F(n+2) + F(n)*F(n+1). - J. M. Bergot, Dec 20 2014
a(n) = 2*F(n)*F(n+1) + (-1)^n. - Bruno Berselli, Oct 30 2015
a(n) = F(2*n+1) - F(n-1)^2 +(-1)^n for n>0. - J. M. Bergot, Jan 19 2016
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
a(n) = F(n+1)^2 + F(n)*F(n-1). See also A099016, tenth formula. - Bruno Berselli, Feb 15 2017
2*a(n) = L(n)*L(n+1) - F(n)*F(n+1), where L = A000032. - Bruno Berselli, Sep 27 2017

A264080 a(n) = 6F(n)F(n+1) + (-1)^n, where F = A000045.

Original entry on oeis.org

1, 5, 13, 35, 91, 239, 625, 1637, 4285, 11219, 29371, 76895, 201313, 527045, 1379821, 3612419, 9457435, 24759887, 64822225, 169706789, 444298141, 1163187635, 3045264763, 7972606655, 20872555201, 54645058949, 143062621645, 374542805987, 980565796315

Offset: 0

Bruno Berselli, Nov 03 2015

a(n) is prime for n = 1, 2, 5, 7, 14, 15, 29, 40, 49, 57, 70, 87, 105, 127, 175, 279, 362, 647, 727, ...

Bruno Berselli, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000032, A000045; A001654, A002878.

Cf. similar sequences of the type k*F(n)*F(n+1)+(-1)^n: A226205 (k=1); A236428 (k=2); A014742 (k=3); A061647 (k=4); A002878 (k=5).

[6*Fibonacci(n)*Fibonacci(n+1)+(-1)^n: n in [0..30]];

a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1,5,13>>)[1, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 28 2016

Table[6 Fibonacci[n] Fibonacci[n + 1] + (-1)^n, {n, 0, 30}]
LinearRecurrence[{2,2,-1},{1,5,13},30] (* Harvey P. Dale, Jul 12 2019 *)

makelist(6*fib(n)*fib(n+1)+(-1)^n, n, 0, 30);

for(n=0, 30, print1(6*fibonacci(n)*fibonacci(n+1)+(-1)^n", "));

a(n) = round((2^(-n)*(-(-2)^n-3*(3-sqrt(5))^n*(-1+sqrt(5))+3*(1+sqrt(5))*(3+sqrt(5))^n))/5) \\ Colin Barker, Sep 28 2016

Vec((1+3*x+x^2)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 28 2016

[6*fibonacci(n)*fibonacci(n+1)+(-1)^n for n in (0..30)]

G.f.: (1+3*x+x^2) / ((1+x)*(1-3*x+x^2)). - Corrected by Colin Barker, Sep 28 2016
a(n) = -a(-n-1) = 2*a(n-1) + 2*a(n-2) - a(n-3) for all n in Z.
a(n) = L(2*n+1) + F(n)*F(n+1) = A002878(n) + A001654(n). See similar identity for A061647.
a(n) = A001654(n+1) + 3*A001654(n) + A001654(n-1).
a(n) - a(n-1) = 2*A099016(n) with a(-1)=-1.
a(n) + a(n-1) = 2*A097134(n) for n>0.
Sum_{i>=0} 1/a(i) = 1.3232560865206157372628688449331...
a(n) = (2^(-n)*(-(-2)^n-3*(3-sqrt(5))^n*(-1+sqrt(5))+3*(1+sqrt(5))*(3+sqrt(5))^n))/5. - Colin Barker, Sep 28 2016
E.g.f.: (1/5)*exp(-x)*(-1 + 6*exp(5*x/2)*(cosh((sqrt(5)*x)/2) + sqrt(5)*sinh((sqrt(5)*x)/2))). - Stefano Spezia, Dec 09 2019

A102714 Expansion of (x+2) / ((x+1)(x^2-3x+1)).

Original entry on oeis.org

2, 5, 14, 36, 95, 248, 650, 1701, 4454, 11660, 30527, 79920, 209234, 547781, 1434110, 3754548, 9829535, 25734056, 67372634, 176383845, 461778902, 1208952860, 3165079679, 8286286176, 21693778850, 56795050373, 148691372270, 389279066436, 1019145827039

Offset: 0

Creighton Dement, Feb 06 2005

A floretion-generated sequence relating Fibonacci numbers.

Floretion Algebra Multiplication Program, FAMP code: (a(n)) = 2dia[I]forseq[ + .5'i + .5'ii' + .5'ij' + .5'ik' ], 2dia[J]forseq = 2dia[K]forseq = A001654, mixforseq = A001519, tesforseq = A099016, vesforseq = A000004. Identity used: dia[I] + dia[J] + dia[K] + mix + tes = ves

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A001519, A099016, A001654, A100545, A055849, A000004.

CoefficientList[Series[(x+2)/((x+1)(x^2-3x+1)),{x,0,30}],x] (* or *) LinearRecurrence[{2,2,-1},{2,5,14},30] (* Harvey P. Dale, Apr 22 2012 *)

a(n) = round((2^(-1-n)*((-1)^n*2^(1+n)+(9-5*sqrt(5))*(3-sqrt(5))^n+(3+sqrt(5))^n*(9+5*sqrt(5))))/5) \\ Colin Barker, Oct 01 2016

Vec((x+2)/((x+1)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Oct 01 2016

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), a(0) = 2, a(1) = 5, a(2) = 14.
a(n) + a(n+1) = A100545(n).
a(n) + 2*a(n+1) + a(n+2) = A055849(n+2).
a(n) + 2*A001654(n) - A099016(n+2) + 2*A001519(n) = 0.
a(n) = (2^(-1-n)*((-1)^n*2^(1+n)+(9-5*sqrt(5))*(3-sqrt(5))^n+(3+sqrt(5))^n*(9+5*sqrt(5))))/5. - Colin Barker, Oct 01 2016
a(n) = (-1)^n +9*A001906(n+1) -A001906(n) . - R. J. Mathar, Sep 11 2019

Corrected by T. D. Noe, Nov 02 2006, Nov 07 2006

cp's OEIS Frontend

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

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A264080 a(n) = 6*F(n)*F(n+1) + (-1)^n, where F = A000045.

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A102714 Expansion of (x+2) / ((x+1)*(x^2-3*x+1)).

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A264080 a(n) = 6F(n)F(n+1) + (-1)^n, where F = A000045.

A102714 Expansion of (x+2) / ((x+1)(x^2-3x+1)).