A273624 -id:A273624 - cp's OEIS frontend

A273623 a(n) = Fibonacci(3n) - (2 + (-1)^n)Fibonacci(n).

1, 5, 32, 135, 605, 2560, 10933, 46305, 196384, 831875, 3524489, 14929920, 63245753, 267913165, 1134902560, 4807524015, 20365009477, 86267563520, 365435291981, 1548008735625, 6557470308896, 27777889982155, 117669030432337, 498454011740160, 2111485077903025

Offset: 1

Peter Bala, May 29 2016

This is a divisibility sequence: if n divides m then a(n) divides a(m). The sequence satisfies a linear recurrence of order 6. In general, for integers r and s, the sequence Fibonacci(r*n) - 2*Fibonacci((r - 2*s)*n) + Fibonacci((r - 4*s)*n) is a divisibility sequence of the sixth order. This is the case r = 3, s = 1. See A127595 (case r = 4, s = 1).

G. C. Greubel, Table of n, a(n) for n = 1..1000
P. Bala, Lucas sequences and divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (4,4,-12,-4,4,1).

Cf. A000032, A000045, A127595, A273622, A273624, A273625.

[Fibonacci(3*n)-(2+(-1)^n)*Fibonacci(n): n in [1..25]]; // Vincenzo Librandi, Jun 02 2016

#A273623
with(combinat):
seq(fibonacci(3*n) - (2 + (-1)^n)*fibonacci(n), n = 1..25);

LinearRecurrence[{4, 4, -12, -4, 4, 1}, {1, 5, 32, 135, 605, 2560}, 100] (* G. C. Greubel, Jun 02 2016 *)
Table[Fibonacci[3 n] - (2 + (-1)^n) Fibonacci[n], {n, 1, 30}] (* Vincenzo Librandi, Jun 02 2016 *)

a(n)=fibonacci(3*n) - (2 + (-1)^n)*fibonacci(n) \\ Charles R Greathouse IV, Jun 08 2016

a(n) = Fibonacci(3*n) - 2*Fibonacci(n) + Fibonacci(-n).
a(2*n) = 5*Fibonacci(2*n)^3;
a(2n+1) = Fibonacci(2*n+1)*(5*Fibonacci(2*n+1)^2 - 4) = Fibonacci(2*n+1)*Lucas(2*n+1)^2.
O.g.f. x*(x^4 - x^3 + 8*x^2 + x + 1)/( (1 + x - x^2 )*(1 - x - x^2)*(1 - 4*x - x^2 ) ).
a(n) = 4*a(n-1) + 4*a(n-2) - 12*a(n-3) - 4*a(n-4) + 4*a(n-5) + a(n-6). - G. C. Greubel, Jun 02 2016

A273625 a(n) = (1/12)(Fibonacci(2n) + Fibonacci(4n) + Fibonacci(6n)).

Original entry on oeis.org

1, 14, 228, 3948, 69905, 1248072, 22352707, 400808856, 7190208684, 129009258070, 2314882621811, 41538234954384, 745368939599413, 13375072472343218, 240005728531700340, 4306726622089196592, 77281063743045412517, 1386752354089549205976, 24884260852952644076119

Offset: 1

Peter Bala, May 31 2016

This is a divisibility sequence, that is, if n divides m then a(n) divides a(m). The sequence satisfies a sixth-order linear recurrence. More generally, the sequence s(n) := Fibonacci(2*n) + Fibonacci(4*n) + ... + Fibonacci(2*k*n) is a divisibility sequence for k = 1,2,3,.... See A215466 for the case k = 2. Cf. A273623, A273624.
From Peter Bala, Aug 05 2019: (Start)
Let  U(n;P,Q), where P and Q are integer parameters, denote the Lucas sequence of the first kind. Then, excluding the cases P = -1 and P = 0, the sequence ( U(n;P,1) + U(2*n;P,1) + U(3*n;P,1))/(P^2 + P) is a sixth-order linear divisibility sequence with o.g.f. x*(1 - 2*(P^2 - 2)*x + (3*P^3 - 3*P^2 - 8*P + 10)*x^2 - 2*(P^2 - 2)*x^3 + x^4)/((1 - P*x + x^2)*(1 - (P^2 - 2)*x + x^2)*(1 - P*(P^2 - 3)*x + x^2)). This is the case P = 3.
More generally, the sequence U(n;P,1) + U(2*n;P,1) + ... + U(k*n;P,1) is a linear divisibility sequence of order 2*k.  See, for example, A215466 with P = 3, k = 2. (End)

G. C. Greubel, Table of n, a(n) for n = 1..795
P. Bala, Lucas sequences and divisibility sequences
P. Bala, Divisibility sequences from strong divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (28,-204,434,-204,28,-1).

Cf. A000045, A001906, A215466, A273623, A273624, A215466.

[1/12*(Fibonacci(2*n)+Fibonacci(4*n)+Fibonacci(6*n)): n in [1..25]]; // Vincenzo Librandi, Jun 02 2016

#A273625
with(combinat):
seq(1/12*(fibonacci(2*n) + fibonacci(4*n) + fibonacci(6*n)), n = 1..20);

LinearRecurrence[{28, -204, 434, -204, 28, -1},{1, 14, 228, 3948, 69905, 1248072}, 100] (* G. C. Greubel, Jun 02 2016 *)
Table[1/12 (Fibonacci[2 n] + Fibonacci[4 n] + Fibonacci[6 n]), {n, 1, 30}] (* Vincenzo Librandi, Jun 02 2016 *)

A001906(n)=fibonacci(2*n)
a(n)=(A001906(n)+A001906(2*n)+A001906(3*n))/12 \\ Charles R Greathouse IV, Jun 08 2016

a(n) = -a(-n).
O.g.f.: x*(x^4 - 14*x^3 + 40*x^2 - 14*x + 1)/((x^2 - 3*x + 1)*(x^2 - 7*x + 1)*(x^2 - 18*x + 1)).
a(n) = 28*a(n-1) - 204*a(n-2) + 434*a(n-3) - 204*a(n-4) + 28*a(n-5) - a(n-6). - G. C. Greubel, Jun 02 2016

A049673 a(n) = (F(3n) + F(n))/3, where F = A000045 (the Fibonacci sequence).

Original entry on oeis.org

0, 1, 3, 12, 49, 205, 864, 3653, 15463, 65484, 277365, 1174889, 4976832, 21082073, 89304891, 378301260, 1602509321, 6788337557, 28755857952, 121811766781, 516002920895, 2185823443596, 9259296684333, 39223010163217, 166151337308544, 703828359351025

Offset: 0

Clark Kimberling

This is an odd divisibility sequence, that is, if n divides m and n/m is odd then a(n) divides a(m). More generally, if r and s are positive integers such that r = s (mod 2) then the sequence Fibonacci(r*n) + Fibonacci(s*n) is an odd divisibility sequence. In the particular case that r is even and s = r + 2 then Fibonacci(r*n) + Fibonacci(s*n) is, in fact, a divisibility sequence. See for example A215466 and A273624. - Peter Bala, May 29 2016

Colin Barker, Table of n, a(n) for n = 0..1000
P. Bala, Lucas sequences and divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (5,-2,-5,-1).

Cf. A000045, A000129, A001045, A215466, A273624.

[(Fibonacci(3*n)+Fibonacci(n))/3: n in [0..30]]; // Vincenzo Librandi, Jun 04 2016

with(combinat): A049673:=n->(fibonacci(3*n)+fibonacci(n))/3: seq(A049673(n), n=0..30); # Wesley Ivan Hurt, Jun 01 2016

Table[(Fibonacci[3 n] + Fibonacci[n])/3, {n, 0, 30}] (* Wesley Ivan Hurt, Jun 01 2016 *)
LinearRecurrence[{5,-2,-5,-1},{0,1,3,12},30] (* Harvey P. Dale, Sep 21 2022 *)

concat(0, Vec(x*(1-2*x-x^2)/((x^2+4*x-1)*(x^2+x-1)) + O(x^30))) \\ Colin Barker, Jun 02 2016

G.f.: x*(1-2*x-x^2) / ((x^2+4*x-1)*(x^2+x-1)). - R. J. Mathar, Oct 26 2015
a(n) = 5*a(n-1) - 2*a(n-2) - 5*a(n-3) - a(n-4) for n>3. - Wesley Ivan Hurt, Jun 01 2016
a(n) = ((-(1/2*(1-sqrt(5)))^n-(2-sqrt(5))^n+(1/2*(1+sqrt(5)))^n+(2+sqrt(5))^n))/(3*sqrt(5)). - Colin Barker, Jun 02 2016
G.f.: G(F(t)), where G(t) is g.f. of A001045 and F(t) is g.f. of A000129. - Oboifeng Dira, Dec 07 2016

cp's OEIS Frontend

A273623 a(n) = Fibonacci(3n) - (2 + (-1)^n)Fibonacci(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A273625 a(n) = (1/12)(Fibonacci(2n) + Fibonacci(4n) + Fibonacci(6n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A049673 a(n) = (F(3n) + F(n))/3, where F = A000045 (the Fibonacci sequence).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

cp's OEIS Frontend

A273623 a(n) = Fibonacci(3*n) - (2 + (-1)^n)*Fibonacci(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A273625 a(n) = (1/12)*(Fibonacci(2*n) + Fibonacci(4*n) + Fibonacci(6*n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A049673 a(n) = (F(3n) + F(n))/3, where F = A000045 (the Fibonacci sequence).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A273623 a(n) = Fibonacci(3n) - (2 + (-1)^n)Fibonacci(n).

A273625 a(n) = (1/12)(Fibonacci(2n) + Fibonacci(4n) + Fibonacci(6n)).