A273625 -id:A273625 - cp's OEIS frontend

A215466 Expansion of x(1-4x+x^2) / ( (x^2-7x+1)(x^2-3*x+1) ).

0, 1, 6, 38, 252, 1705, 11628, 79547, 544824, 3733234, 25585230, 175356611, 1201893336, 8237850373, 56462937882, 387002396990, 2652553009008, 18180866487757, 124613506702404, 854113665498719, 5854182112700460

Offset: 0

R. J. Mathar, Aug 11 2012

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 case P = -1, the sequence ( U(n;P,1) + U(2*n;P,1) )/(P + 1) is a fourth-order linear divisibility sequence with o.g.f. x*(1 - 2*(P - 1)*x + x^2)/((1 - P*x + x^2)*(1 - (P^2 - 2)*x + x^2)). This is the case P = 3. See A000027 (P = 2), A165998 (P = -2) and A238536 (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. As an example, see A273625 (P = 3, k = 3 and then sequence normalized with initial term 1). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Lucas sequence
Peter Bala, Divisibility sequences from strong divisibility sequences
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.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Odd and even linear divisibility sequences of order 4, INTEGERS, 2015, #A33.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (10,-23,10,-1).

Cf. A000032, A000045, A001906, A033888, A165998, A215465, A238536, A273625.

I:=[0,1,6,38]; [n le 4 select I[n] else 10*Self(n-1)-23*Self(n-2)+10*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 23 2012

/* By definition: */ m:=20; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1-4*x+x^2)/((x^2-7*x+1)*(x^2-3*x+1)))); // Bruno Berselli, Dec 24 2012

A215466 := proc(n)
    if type(n,'even') then
        A000032(n)*combinat[fibonacci](3*n)/4 ;
    else
        combinat[fibonacci](n)*A000032(3*n)/4 ;
    end if;
end proc:

CoefficientList[Series[x*(1 - 4*x + x^2)/((x^2 - 7*x + 1)*(x^2 - 3*x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 23 2012 *)
LinearRecurrence[{10,-23,10,-1},{0,1,6,38},30] (* Harvey P. Dale, Nov 02 2015 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,10,-23,10]^n*[0;1;6;38])[1,1] \\ Charles R Greathouse IV, Nov 13 2015

{a(n) = my(w=quadgen(5)^(2*n)); imag(w^2+w)/4}; /* Michael Somos, Dec 29 2022 */

a(n) = L(n)*F(3n)/4 if n even, = F(n)*L(3n)/4 if n odd, where L=A000032, F=A000045.
a(n) = 3*A004187(n)/4 + A001906(n)/4.
a(n) = 10*a(n-1) - 23*a(n-2) + 10*a(n-3) - a(n-4), a(0)=0, a(1)=1, a(2)=6, a(3)=38. - Harvey P. Dale, Nov 02 2015
a(n) = (1/4)*(Fibonacci(2*n) + Fibonacci(4*n)) = (1/4)*(A001906(n) + A033888(n)). - Peter Bala, Aug 05 2019
E.g.f.: exp(5*x/2)*(cosh(x)+exp(x)*cosh(sqrt(5)*x))*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Aug 17 2019
a(n) = -a(-n) for all n in Z. - Michael Somos, Dec 29 2022

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

A273624 a(n) = (1/11)(Fibonacci(4n) + Fibonacci(6*n)).

Original entry on oeis.org

1, 15, 248, 4305, 76255, 1361520, 24384737, 437245935, 7843863784, 140737371825, 2525326494911, 45314438127840, 813129752279233, 14590988151618255, 261824431125415640, 4698247224097107345, 84306614992412658847, 1512820749915870503760, 27146466385039244529569

Offset: 1

Peter Bala, May 29 2016

This is a divisibility sequence: if n divides m then a(n) divides a(m). More generally, if r is an even integer then the sequence Fibonacci(r*n) + Fibonacci((r + 2)*n) is a divisibility sequence. See A215466 for the case r = 2.

Also, the sequence s(n) := Fibonacci(4*n) + Fibonacci(6*n) + ... + Fibonacci((2*k + 2)*n) is a divisibility sequence when k is a positive integer that is not a multiple of 3.

G. C. Greubel, Table of n, a(n) for n = 1..795
P. Bala, Lucas sequences and divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (25,-128,25,-1).

Cf. A000045, A049673, A215466, A273623, A273625.

[1/11*(Fibonacci(4*n)+Fibonacci(6*n)): n in [1..25]]; // Vincenzo Librandi, Jun 02 2016

#A273624
with(combinat):
seq(1/11*(fibonacci(4n) + fibonacci(6n)), n = 1..20);

LinearRecurrence[{25,-128,25,-1},{1, 15, 248, 4305},100] (* G. C. Greubel, Jun 02 2016 *)
Table[1/11 (Fibonacci[4 n] + Fibonacci[6 n]), {n, 1, 30}] (* Vincenzo Librandi, Jun 02 2016 *)

a(n)=(fibonacci(4*n) + fibonacci(6*n))/11 \\ Charles R Greathouse IV, Jun 08 2016

a(n) = -a(-n).
a(n) = 25*a(n-1) - 128*a(n-2) + 25*a(n-3) - a(n-4).
O.g.f. (x^2 - 10*x + 1)/((x^2 - 7*x + 1)*(x^2 - 18*x + 1)).

cp's OEIS Frontend

A215466 Expansion of x*(1-4*x+x^2) / ( (x^2-7*x+1)*(x^2-3*x+1) ).

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

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A273624 a(n) = (1/11)*(Fibonacci(4*n) + Fibonacci(6*n)).

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A215466 Expansion of x(1-4x+x^2) / ( (x^2-7x+1)(x^2-3*x+1) ).

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

A273624 a(n) = (1/11)(Fibonacci(4n) + Fibonacci(6*n)).