A127595 -id:A127595 - cp's OEIS frontend

A238536 A fourth-order linear divisibility sequence related to the Fibonacci numbers: a(n) = (1/2)Fibonacci(3n)*Lucas(n).

1, 12, 68, 504, 3355, 23256, 158717, 1089648, 7463884, 51170460, 350695511, 2403786672, 16475579353, 112925875764, 774003961940, 5305106018016, 36361727272627, 249227013404808, 1708227291909269, 11708364225400920, 80250321774226396, 550043889533755332, 3770056901455017263

Offset: 1

Peter Bala, Feb 28 2014

Let P and Q be integers. The Lucas sequences U(n) and V(n) (which depend on P and Q) are a pair of integer sequences that satisfy the recurrence equation a(n) = P*a(n-1) - Q*a(n-2) with the initial conditions U(0) = 0, U(1) = 1 and V(0) = 2, V(1) = P, respectively. The sequence {U(n)} n >= 1 is a linear divisibility sequence of order 2, i.e., U(n) divides U(m) whenever n divides m and U(n) <> 0. In general, V(n) is not a divisibility sequence. However, it can be shown that if p >= 3 is an odd integer then the sequence {U(p*n)*V(n)} n >= 1 is a linear divisibility sequence of order 4. For a proof and a generalization of this result see the Bala link. Here we take p = 3 with P = 1 and Q = -1, for which U(n) is the sequence of Fibonacci numbers, A000045, V(n) is the sequence of Lucas numbers, A000032, and normalize the sequence to have the initial term 1. For other sequences of this type see A238537 and A238538.

S. Koshkin, Non-classical linear divisibility sequences ..., Fib. Q., 57 (No. 1, 2019), 68-80.

G. C. Greubel, Table of n, a(n) for n = 1..1185
Peter Bala, A family of linear divisibility sequences of order four
E. L. Roettger and H. C. Williams, Appearance of Primes in Fourth-Order Odd Divisibility Sequences, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
Wikipedia, Divisibility sequence
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (4,19,4,-1).

Cf. A000032, A000045, A127595, A215466, A238537, A238538.

I:=[12, 68, 504, 3355]; [1] cat [n le 4 select I[n] else 4*Self(n-1) + 19*Self(n-2) + 4*Self(n-3) - Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 25 2017

with(combinat): lucas:= n->fibonacci(n+1)+ fibonacci(n-1):
seq(1/2*lucas(n)*fibonacci(3*n), n = 1..24);

Table[Fibonacci(3*n)*Lucas(n)/2, {n,1,30}] (* or *) Join[{1}, LinearRecurrence[{4,19,4,-1}, {12, 68, 504, 3355}, 30]] (* G. C. Greubel, Dec 25 2017 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,4,19,4]^(n-1)*[1;12;68;504])[1,1] \\ Charles R Greathouse IV, Oct 07 2016

a(n) = (1/2)*Fibonacci(3*n)*Lucas(n) = (1/2)*A000045(3*n)*A000032(n).
a(n) = (1/2)*Fibonacci(2*n)*Fibonacci(3*n)/Fibonacci(n).
a(n) = (1/(2*sqrt(5)))*( ((7 + 3*sqrt(5))/2)^n - ((7 - 3*sqrt(5))/2)^n + (-1)^n*((3 + sqrt(5))/2)^n - (-1)^n*((3 - sqrt(5))/2)^n ).
The sequence can be extended to negative indices by setting a(-n) = -a(n).
O.g.f. x*(1 + 8*x + x^2)/( (1 + 3*x + x^2)*(1 - 7*x + x^2) ).
Recurrence equation: a(n) = 4*a(n-1) + 19*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = (1/2) * (Fibonacci(4*n) + (-1)^n*Fibonacci(2*n)). - Ralf Stephan, Mar 01 2014

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

Original entry on oeis.org

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

A273626 A fourth-order divisibility sequence: a(n) = (1/14)(Pell(4n) + Pell(2*n)).

Original entry on oeis.org

1, 30, 995, 33660, 1142629, 38810970, 1318402631, 44786716920, 1521429030985, 51683794848150, 1755727563817259, 59643053188493940, 2026108079758297261, 68828031652259981010, 2338126968060165944975, 79427488882178225107440, 2698196495024745460575889

Offset: 1

Peter Bala, May 31 2016

This is a divisibility sequence, that is, a(n) divides a(m) whenever n divides m. The sequence satisfies a linear recurrence of order 4.

A000129(n) = Pell(n) is the Lucas sequence U_n(2,-1). In general, if U(n) = U_n(P,Q) is the Lucas sequence with integer parameters P and Q then when Q = 1 or Q = -1 both U(4*n) + U(2*n) and U(4*n) - 2*U(2*n) are divisibility sequences of the fourth order. Cf. A127595, A215466 and A273627.

Colin Barker, Table of n, a(n) for n = 1..600
P. Bala, Lucas sequences and divisibility sequences
Wikipedia, Lucas Sequence
Index entries for linear recurrences with constant coefficients, signature (40,-206,40,-1).

Cf. A000129, A001109, A029547, A081555, A127595, A215466, A273627.

I:=[1,30,995,33660]; [n le 4 select I[n] else 40*Self(n-1)-206*Self(n-2)+40*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 04 2016

#A273626
A000129 := proc (n) option remember;
if n <= 1 then n else 2*A000129(n-1) + A000129(n-2) end if
end proc:
seq(1/14*(A000129(4*n) + A000129(2*n)), n = 1..20);

LinearRecurrence[{40,-206,40,-1},{1,30,995,33660},100] (* G. C. Greubel, Jun 02 2016 *)

a(n) = sqrt(2)/56*( (sqrt(2) + 1)^(4*n) - (sqrt(2) - 1)^(4*n) + (sqrt(2) + 1)^(2*n) - (sqrt(2) - 1)^(2*n) ).
a(n) = (A082405(n) + A001109(n))/7 .
a(n) = 1/14*Pell(2*n)*A081555(n).
a(n) = -a(-n).
a(n) = 40*a(n-1) - 206*a(n-2) + 40*a(n-3) - a(n-4) for n>4.
O.g.f.: x*(x^2 - 10*x + 1)/((x^2 - 6*x + 1)*(x^2 - 34*x + 1)).

A273627 A divisibility sequence: (1/8)(Pell(4n) - 2Pell(2n)).

Original entry on oeis.org

1, 48, 1715, 58752, 1998709, 67914000, 2307174311, 78376578048, 2662499775145, 90446634986352, 3072523201721819, 104375342876112000, 3545689138389464221, 120449055384533383248, 4091722194064948458575, 138998105543576763850752, 4721843866291934117429329

Offset: 1

Peter Bala, May 30 2016

The sequence of Pell numbers A000129 is a well-known divisibility sequence of order 2, that is, the sequence satisfies a linear recurrence of order 2 and Pell(n) divides Pell(m) whenever n divides m. The linear combinations Pell(4*n) - 2*Pell(2*n) and Pell(4*n) + Pell(2*n) are also divisibility sequences, this time of order 4. Cf. A127595 and A273626.

Colin Barker, Table of n, a(n) for n = 1..600
P. Bala, Lucas sequences and divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (40,-206,40,-1).

Cf. A000129, A002203, A127595, A273626.

I:=[1,48,1715,58752]; [n le 4 select I[n] else 40*Self(n-1)-206*Self(n-2)+40*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 02 2016

#A273627
A000129 := proc (n) option remember;
if n <= 1 then n else 2*A000129(n-1)+A000129(n-2) end if
end proc:
seq(1/8*(A000129(4*n)-2*A000129(2*n)), n = 1..20);

CoefficientList[Series[(x^2 + 8*x + 1)/((x^2 - 6*x + 1)*(x^2 - 34*x + 1)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jun 01 2016 *)
LinearRecurrence[{40, -206, 40, -1}, {1, 48, 1715, 58752}, 20] (* Vincenzo Librandi, Jun 02 2016 *)

a(n) = A000129(4*n) - 2*A000129(2*n).
a(n) = A000129(2*n)*(A002203(2*n) - 2).
a(n) = -a(-n).
a(n) = 40*a(n-1) - 206*a(n-2) + 40*a(n-3) - a(n-4) for n>4.
O.g.f.: x*(x^2 + 8*x + 1)/((x^2 - 6*x + 1)*(x^2 - 34*x + 1)).
a(n) = ((17+12*sqrt(2))^(1-n)*(-1+2*(3+2*sqrt(2))^n+(17+12*sqrt(2))^(2*n)-2*(99+70*sqrt(2))^n))/(384+272*sqrt(2)). - Colin Barker, Jun 02 2016

cp's OEIS Frontend

A238536 A fourth-order linear divisibility sequence related to the Fibonacci numbers: a(n) = (1/2)*Fibonacci(3*n)*Lucas(n).

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

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A273626 A fourth-order divisibility sequence: a(n) = (1/14)*(Pell(4*n) + Pell(2*n)).

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A273627 A divisibility sequence: (1/8)*(Pell(4*n) - 2*Pell(2*n)).

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A238536 A fourth-order linear divisibility sequence related to the Fibonacci numbers: a(n) = (1/2)Fibonacci(3n)*Lucas(n).

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

A273626 A fourth-order divisibility sequence: a(n) = (1/14)(Pell(4n) + Pell(2*n)).

A273627 A divisibility sequence: (1/8)(Pell(4n) - 2Pell(2n)).