A215465 -id:A215465 - 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

A273622 a(n) = (1/3)(Lucas(3n) - Lucas(n)).

Original entry on oeis.org

1, 5, 24, 105, 451, 1920, 8149, 34545, 146376, 620125, 2626999, 11128320, 47140601, 199691245, 845906424, 3583318305, 15179181851, 64300049280, 272379384749, 1153817597625, 4887649790376, 20704416783605, 87705316964399, 371525684705280, 1573808055889201, 6666757908429845

Offset: 1

Peter Bala, May 27 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. Cf. A273623.

More generally, for distinct integers r and s with r == s (mod 2), the sequence Lucas(r*n) - Lucas(s*n) is a fourth-order divisibility sequence. When r is even (resp. odd) the normalized sequence (Lucas(r*n) - Lucas(s*n))/(Lucas(r) - Lucas(s)), with initial term equal to 1, has the o.g.f. x*(1 - x^2)/( (1 - Lucas(r)*x + x^2)*(1 - Lucas(s)*x + x^2) ) (resp. x*(1 + x^2)/( (1 - Lucas(r)*x - x^2)*(1 - Lucas(s)*x - x^2) )) and belongs to the 3-parameter family of fourth-order divisibility sequences found by Williams and Guy, with parameter values P1 = (Lucas(r) + Lucas(s)), P2 = Lucas(r)*Lucas(s) and Q = 1 (resp. Q = -1). For particular cases see A004146 (r = 2, s = 0), A049684 (r = 4, s = 0), A215465 (r = 4, s = 2), A049683 (r = 6, s = 0), A049682 (r = 8, s = 0) and A037451 (r = 3, s = -1).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Peter Bala, Lucas sequences and divisibility sequences
Spirit Karcher and Mariah Michael, Prime Factors and Divisibility of Sums of Powers of Fibonacci and Lucas Numbers, Amer. J. of Undergraduate Research (2021) Vol. 17, Issue 4, 59-69.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory, 7 (5) (2011), 1255-1277.
Index entries for linear recurrences with constant coefficients, signature (5,-2,-5,-1).

Cf. A000032, A037451, A004146, A049682, A049683, A049684, A215465, A273623.

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

#A273622
with(combinat):
Lucas := n->fibonacci(n+1) + fibonacci(n-1):
seq(1/3*(Lucas(3*n) - Lucas(n)), n = 1..24);

LinearRecurrence[{5,-2,-5,-1}, {1, 5, 24, 105}, 100] (* G. C. Greubel, Jun 02 2016 *)
Table[1/3 (LucasL[3 n] - LucasL[n]), {n, 1, 30}] (* Vincenzo Librandi, Jun 02 2016 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,-5,-2,5]^(n-1)*[1;5;24;105])[1,1] \\ Charles R Greathouse IV, Jun 07 2016

a(n) = (1/3)*( (2 + sqrt(5))^n + (2 - sqrt(5))^n - ((1 + sqrt(5))/2)^n - ((1 - sqrt(5))/2)^n ).
a(n) = -a(-n).
a(n) = 5*a(n-1) - 2*a(n-2) - 5*a(n-3) - a(n-4).
O.g.f.: x*(1 + x^2)/((1 - x - x^2 )*(1 - 4*x - x^2)).
a(n) = (A014448(n) - A000032(n))/3. - R. J. Mathar, Jun 07 2016
a(n) = Fibonacci(n) + Sum_{k=1..n} Fibonacci(n-k)*Lucas(3*k). - Yomna Bakr and Greg Dresden, Jun 16 2024
E.g.f.: (2*exp(2*x)*cosh(sqrt(5)*x) - 2*exp(x/2)*cosh(sqrt(5)*x/2))/3. - Stefano Spezia, Jun 17 2024

A085695 a(n) = Fibonacci(n)*Fibonacci(3n)/2.

Original entry on oeis.org

0, 1, 4, 34, 216, 1525, 10336, 71149, 486864, 3339106, 22881100, 156843721, 1074985344, 7368157369, 50501844796, 346145466850, 2372514562656, 16261461342589, 111457702083424, 763942486626661, 5236139616899400

Offset: 0

Emanuele Munarini, Jul 18 2003

This is a divisibility sequence, that is, if n | m then a(n) | a(m). However, it is not a strong divisibility sequence. It is the case k = -3 of a 1-parameter family of 4th-order linear divisibility sequences with o.g.f. x*(1 - x^2)/( (1 - k*x + x^2)*(1 - (k^2 - 2)*x + x^2) ). Compare with A000290 (case k = 2) and A215465 (case k = 3). - Peter Bala, Jan 17 2014

a(n) + a(n+1) is the numerator of the continued fraction [1,...,1,4,...,4] with n 1's followed by n 4's. - Greg Dresden and Hexuan Wang, Aug 16 2021

			G.f. = x + 4*x^2 + 34*x^3 + 216*x^4 + 1525*x^5 + 10336*x^6 + ... - _Michael Somos_, Apr 23 2022

Michael De Vlieger, Table of n, a(n) for n = 0..1197
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
Index entries for linear recurrences with constant coefficients, signature (4,19,4,-1).

Cf. A153173, A215465.

Array[Times @@ MapIndexed[Fibonacci[#]/First@ #2 &, {#, 3 #}] &, 21, 0] (* or *) LinearRecurrence[{4, 19, 4, -1}, {0, 1, 4, 34}, 21] (* or *)
CoefficientList[Series[(x - x^3)/(1 - 4 x - 19 x^2 - 4 x^3 + x^4), {x, 0, 20}], x] (* Michael De Vlieger, Dec 17 2017 *)

numlib::fibonacci(3*n)*numlib::fibonacci(n)/2 $ n = 0..35; // Zerinvary Lajos, May 13 2008

a(n) = fibonacci(n)*fibonacci(3*n)/2 \\ Andrew Howroyd, Dec 17 2017

G.f.: ( x - x^3 )/( 1 - 4 x - 19 x^2 - 4 x^3 + x^4 ).
Recurrence: a(n+4) = 4*a(n+3) + 19*a(n+2) + 4*a(n+1) - a(n).
a(n) = a(-n) and A153173(n) = 1 + 10*a(n) for all n in Z. - Michael Somos, Apr 23 2022

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

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A273622 a(n) = (1/3)*(Lucas(3*n) - Lucas(n)).

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A085695 a(n) = Fibonacci(n)*Fibonacci(3n)/2.

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

A273622 a(n) = (1/3)(Lucas(3n) - Lucas(n)).