A238536 -id:A238536 - 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

A238537 A fourth-order linear divisibility sequence related to the Pell numbers.

Original entry on oeis.org

1, 42, 1379, 47124, 1599205, 54335358, 1845747527, 62701403688, 2130000094537, 72357312787410, 2458018570699691, 83500274463891516, 2836551311028252973, 96359244313163973414, 3273377755262716618895, 111198484435049515150416, 3777475093033912744231057

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 divisibility sequence, i.e., U(n) divides U(m) whenever n divides m and U(n) <> 0. In general the sequence 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 divisibility sequence satisfying a linear recurrence of order 4. For a proof and a generalization of this result see the Bala link. Here we take p = 3 with P = 2 and Q = -1, for which U(n) is the sequence of Pell numbers A000129, and consider the normalized divisibility sequence with initial term equal to 1. For other sequences of this type see A238536 and A238538

Michael De Vlieger, Table of n, a(n) for n = 1..654
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, Pell number
Index entries for linear recurrences with constant coefficients, signature (28,202,28,-1)

Cf. A000032, A000045, A215466, A238536, A238538, A238539, A238540, A238541.

LinearRecurrence[{28, 202, 28, -1}, {1, 42, 1379, 47124}, 17] (* Jean-François Alcover, Nov 02 2019 *)

a(n) = (1/5)*A000129(3*n)*A001333(n).
a(n) = (1/(20*sqrt(2)))*((1 + sqrt(2))^(3*n) - (1 - sqrt(2))^(3*n))*( (1 + sqrt(2))^n + (1 - sqrt(2))^n ).
O.g.f.: x*(1 + 14*x + x^2)/( (1 + 6*x + x^2)*(1 - 34*x + x^2) ).
Recurrence equation: a(n) = 28*a(n-1) + 202*a(n-2) + 28*a(n-3) - a(n-4).
a(n) = (1/10) * (Pell(4n) + (-1)^n*Pell(2n)), with Pell(n) = A000129(n). - Ralf Stephan, Mar 01 2014

A238538 A fourth-order linear divisibility sequence: a(n) = (2^n + 1)(2^(3n) - 1)/ ( (2 + 1)*(2^3 - 1) ).

Original entry on oeis.org

1, 15, 219, 3315, 51491, 811395, 12882499, 205321155, 3278747331, 52408827075, 838132189379, 13406842675395, 214483303960771, 3431523432591555, 54902699475185859, 878429788032676035, 14054769379960303811, 224875452250864496835, 3598000373385828511939

Offset: 1

Peter Bala, Feb 28 2014

This is a fourth-order linear divisibility sequence, that is, the sequence satisfies a linear recurrence of order 4 and if n | m then a(n) | a(m). This is a consequence of the following more general result: The polynomials P(n,x,y) := (x^n + y^n)*(x^(3*n) - y^(3*n)) form a fourth-order linear divisibility sequence in the polynomial ring Z[x,y]. See the Bala link.
Hence, for a fixed integers M and N, the normalized sequence (M^n + N^n)*(M^(3*n) - N^(3*n))/ ( (M + N)*(M^3 - N^3) ) for n = 1,2,3,... is a linear divisibility sequence of order 4. It has the rational o.g.f. x*(1 - 2*M*N*(M^2 - M*N + N^2)*x + (M*N)^4*x^2)/( (1 - M^4*x)*(1 - M^3*N*x)*(1 - M*N^3*x)*(1 - N^4*x) ). This is the case M = 2, N = 1. For other cases see A238539(M = 2, N = -1), A238540(M = 3, N = 1) and A238541(M = 3, N = 2). See also A238536, A238537 and A215466.
Note, these sequences do not belong to the family of linear divisibility sequences of the fourth order studied by Williams and Guy, which have o.g.f.s of the form x*(1 - q*x^2)/Q(x), Q(x) a quartic polynomial and q an integer parameter.

Michael De Vlieger, Table of n, a(n) for n = 1..831
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
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, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (27,-202,432,-256).

Cf. A215466, A236536, A236537, A238539, A238540, A238541.

seq(1/21*(2^n + 1)*(2^(3*n) - 1), n = 1..20);

LinearRecurrence[{27,-202,432,-256},{1,15,219,3315},20] (* Harvey P. Dale, Jul 04 2019 *)

a(n) = (1/21)*(2^n + 1)*(2^(3*n) - 1) = A000051(n)*A024088(n)/21.
a(n) = (1/21)*(4^n - 1)*(8^n - 1)/(2^n - 1).
O.g.f.: x*(1 - 12*x + 16*x^2)/((1 - x)*(1 - 2*x)*(1 - 8*x)*(1 - 16*x)).
Recurrence equation: a(n) = 27*a(n-1) - 202*a(n-2) + 432*a(n-4) - 256*a(n-4).

A238600 A sixth-order linear divisibility sequence related to the Fibonacci numbers: a(n) := (1/6)Fibonacci(3n)Fibonacci(4n)/Fibonacci(n).

Original entry on oeis.org

1, 28, 408, 7896, 137555, 2496144, 44599477, 801617712, 14375440584, 258018516140, 4629531440711, 83076469908768, 1490726895438793, 26750144944686436, 480010941060482040, 8613453244178393184, 154562103244937408987, 2773504708179098411952

Offset: 1

Peter Bala, Mar 01 2014

Let P and Q be relatively prime integers. The Lucas sequence U(n) (which depends on P and Q) is an integer sequence that satisfies the recurrence equation a(n) = P*a(n-1) - Q*a(n-2) with the initial conditions U(0) = 0, U(1) = 1. The sequence {U(n)}n>=1 is a strong divisibility sequence, i.e., gcd(U(n),U(m)) = |U(gcd(n,m))|. It follows that {U(n)} is a divisibility sequence, i.e., U(n) divides U(m) whenever n divides m and U(n) <> 0.
It can be shown that if p and q are a pair of relatively prime positive integers, and if U(n) never vanishes, then the sequence {U(p*n)*U(q*n)/U(n)}n>=1 is a linear divisibility sequence of order 2*min(p,q). For a proof and a generalization of this result see the Bala link.
Here we take p = 3 and q = 4 with P = 1 and Q = -1, for which U(n) is the sequence of Fibonacci numbers, A000045, and normalize the sequence to have the initial term 1.
For other sequences of this type see A238601, A238602 and A238603. See also A238536.

G. C. Greubel, Table of n, a(n) for n = 1..500
P. Bala, Divisibility sequences from strong divisibility sequences
Wikipedia, Divisibility sequence
Wikipedia, Fibonacci number
Wikipedia, Lucas Sequence
Index entries for linear recurrences with constant coefficients, signature (14,90,-350,90,14,-1).

Cf. A000045, A215466, A238536, A238601, A238602, A238603.

[(Fibonacci(2*n) + (-1)^n*Fibonacci(4*n) + Fibonacci(6*n))/6: n in [1..30]]; // G. C. Greubel, Aug 07 2018

with(combinat):
seq(1/6*fibonacci(3*n)*fibonacci(4*n)/fibonacci(n), n = 1..20);

Table[(1/6)*(Fibonacci[2*n] + (-1)^n*Fibonacci[4*n] + Fibonacci[6*n]), {n, 1, 500}] (* G. C. Greubel, Aug 07 2018 *)
LinearRecurrence[{14,90,-350,90,14,-1},{1,28,408,7896,137555,2496144},20] (* Harvey P. Dale, Aug 26 2020 *)

vector(30, n, (fibonacci(2*n) + (-1)^n*fibonacci(4*n) + fibonacci(6*n))/6) \\ G. C. Greubel, Aug 07 2018

a(n) = (1/6)*Fibonacci(3*n)*Fibonacci(4*n)/Fibonacci(n).
a(n) = (1/6)*( Fibonacci(2*n) + (-1)^n*Fibonacci(4*n) + Fibonacci(6*n) ).
The sequence can be extended to negative indices when a(-n) = -a(n).
O.g.f. x*(1 + 14*x - 74*x^2 + 14*x^3 + x^4)/( (1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2) ).
Recurrence equation: a(n) = 14*a(n-1) + 90*a(n-2) - 350*a(n-3) + 90*a(n-4) + 14*a(n-5) - a(n-6).

A238540 A fourth-order linear divisibility sequence: a(n) := (3^n + 1)(3^(3n) - 1)/( (3 + 1)*(3^3 - 1)).

Original entry on oeis.org

1, 70, 5299, 419020, 33664741, 2719393810, 220069738519, 17820217484440, 1443290970139081, 116902609136432350, 9469004435040169339, 766986472802959676260, 62125826363286791503021, 5032189831214900660779690, 407607319514701058318401759, 33016191346720726553176114480

Offset: 1

Peter Bala, Mar 01 2014

This is a divisibility sequence, that is, if n | m then a(n) | a(m). More generally, the polynomials P(n,x) := (x^n + 1)*(x^(3*n) - 1) form a sequence of divisibility polynomials in the polynomial ring Z[x]; that is, if n divides m then P(n,x) divides P(m,x) in Z[x]. See the Bala link for a proof and generalization. Here we consider the integer sequence coming from the normalized polynomials P(n,x)/P(n,1) at x = 3.
The sequence satisfies a homogeneous linear recurrence of the fourth order. However, it does not belong to the family of linear divisibility sequences of the fourth order discovered by Williams and Guy, which have o.g.f.s of the form x*(1 - q*x^2)/Q(x), Q(x) a quartic polynomial and q an integer parameter.
For sequences of a similar type see A238536 through A238541.

Peter Bala, A family of linear divisibility sequences of order four
Wikipedia, Divisibility sequence
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, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (112,-2622,9072,-6561).

Cf. A238536, A238537, A238538, A238539, A238541.

#A238540
seq(1/104*(3^n + 1)*(3^(3*n) - 1), n = 1..20);

LinearRecurrence[{112, -2622, 9072, -6561}, {1, 70, 5299, 419020}, 16] (* Jean-François Alcover, Nov 14 2019 *)

a(n) = (1/104)*(3^n + 1)*(3^(3*n) - 1) = (1/104)*(9^n - 1)*(27^n - 1)/(3^n - 1).
O.g.f.: x*(1 - 42*x + 81*x^2)/((1 - x)*(1 - 3*x)*(1 - 27*x)*(1 - 81*x)).
Recurrence equation: a(n) = 112*a(n-1) - 2622*a(n-2) + 9072*a(n-3) - 6561*a(n-4).

A238541 A fourth-order linear divisibility sequence: a(n) := A(n)/A(1) where A(n) := ( (3^n + 2^n)(3^(3n) - 2^(3*n)) ).

Original entry on oeis.org

1, 91, 7063, 538447, 41441455, 3231753343, 254851186927, 20265345051679, 1621012954550479, 130194036583465855, 10485834936321976111, 846117830539227426271, 68360837263665964839823, 5527792975131721247371327, 447241733557623755497669615

Offset: 1

Peter Bala, Mar 01 2014

The bivariate polynomials P(n,x,y) := (x^n + y^n)*(x^(3*n) - y^(3*n)) form a sequence of divisibility polynomials in the polynomial ring Z[x,y]; that is, if n divides m then P(n,x,y) divides P(m,x,y) in Z[x,y] (see the Bala link). Here we consider the integer sequence coming from the normalized polynomials P(n,x,y)/P(1,x,y) when x = 3 and y = 2. Other cases include A238538(x = 2, y = 1), A238539(x = 2, y = -1) and A238540(x = 3, y = 1). See also A238536, A238537 and A215466.

Peter Bala, A family of linear divisibility sequences of order four
Wikipedia, Divisibility sequence
Index entries for linear recurrences with constant coefficients, signature (175,-10158,226800,-1679616).

Cf. A215466, A238536, A238537, A238538, A238539, A238540.

#A238541
seq(1/95*(3^n + 2^n)*(3^(3*n) - 2^(2*n)), n = 1..20);

LinearRecurrence[{175,-10158,226800,-1679616},{1,91,7063,538447},20] (* Harvey P. Dale, Apr 12 2018 *)

a(n) = (1/95)*(3^n + 2^n)*(3^(3*n) - 2^(3*n)).
a(n) = (1/95)*(9^n - 4^n)*(27^n - 8^n)/(3^n - 2^n).
O.g.f.: x*(1 - 84*x + 1296*x^2)/((1 - 16*x)*(1 - 24*x)*(1 - 54*x)*(1 - 81*x)).
Recurrence equation: a(n) = 175*a(n-1) - 10158*a(n-2) + 226800*a(n-4) - 1679616*a(n-4).

A238539 A fourth-order linear divisibility sequence: a(n) := (1/9)(2^n + (-1)^n)(2^(3*n) - (-1)^n).

Original entry on oeis.org

1, 35, 399, 7735, 112871, 1893255, 29593159, 479082695, 7620584391, 122287263175, 1953732901319, 31282632909255, 500338874618311, 8006888009380295, 128098480026087879, 2049669505409577415, 32793961486615474631, 524709388585350492615, 8395302178969583120839

Offset: 1

Peter Bala, Mar 01 2014

This is a divisibility sequence, that is, if n | m then a(n) | a(m). This is a consequence of the following more general result: The polynomials P(n,x,y) := (x^n + y^n)*(x^(3*n) - y^(3*n)) form a divisibility sequence in the polynomial ring Z[x,y]. See the Bala link.
The sequence satisfies a homogeneous linear recurrence of the fourth order. However, it does not belong to the family of linear divisibility sequences of the fourth order studied by Williams and Guy, which have o.g.f.s of the form x*(1 - q*x^2)/Q(x), Q(x) a quartic polynomial and q an integer.
For sequences of a similar type see A238536 through A238541.

Peter Bala, A family of linear divisibility sequences of order four
Wikipedia, Divisibility sequence"
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, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (7,138,112,-256).

Cf. A238536, A238537, A238538, A238540, A238541.

seq(1/9*(2^n + (-1)^n)*(2^(3*n) - (-1)^n), n = 1..20);

a(n) = (1/9)*(2^n + (-1)^n)*(2^(3*n) - (-1)^n) = (1/9)*(4^n - 1)*(8^n - (-1)^n)/(2^n - (-1)^n).
O.g.f.: x*(1 + 28*x + 16*x^2)/((1 - x)*(1 + 2*x)*(1 + 8*x)*(1 - 16*x)).
Recurrence equation: a(n) = 7*a(n-1) + 138*a(n-2) + 112*a(n-4) - 256*a(n-4).

A238601 A sixth-order linear divisibility sequence related to the Fibonacci numbers: a(n) := (1/10)Fibonacci(3n)Fibonacci(5n)/Fibonacci(n).

Original entry on oeis.org

1, 44, 1037, 32472, 915305, 26874892, 776952553, 22595381424, 655633561309, 19040507781020, 552780012054689, 16050219184005336, 466002944275859873, 13530204273746536948, 392841165312292809085, 11405932444267712654688, 331164788382150547106857, 9615185834308570310716196

Offset: 1

Peter Bala, Mar 06 2014

Let P and Q be relatively prime integers. The Lucas sequence U(n) (which depends on P and Q) is an integer sequence that satisfies the recurrence equation a(n) = P*a(n-1) - Q*a(n-2) with the initial conditions U(0) = 0, U(1) = 1. The sequence {U(n)}n>=1 is a strong divisibility sequence, i.e., gcd(U(n),U(m)) = |U(gcd(n,m))|. It follows that {U(n)} is a divisibility sequence, i.e., U(n) divides U(m) whenever n divides m and U(n) <> 0.
It can be shown that if p and q are a pair of relatively prime positive integers, and if U(n) never vanishes, then the sequence {U(p*n)*U(q*n)/U(n)}n>=1 is a linear divisibility sequence of order 2*min(p,q). For a proof and a generalization of this result see the Bala link.
Here we take p = 3 and q = 5 with P = 1 and Q = -1, for which U(n) is the sequence of Fibonacci numbers, A000045, and normalize the sequence {U(3*n)*U(5*n)/U(n)}n>=1 to have the initial term 1.
For other sequences of this type see A238600, A238602 and A238603. See also A238536.
Since Fibonacci(n) can be defined for all n, so can this sequence. - N. J. A. Sloane, May 07 2017

			G.f. = x + 44*x^2 + 1037*x^3 + 32472*x^4 + 915305*x^5 + 26874892*x^6 + ... - _Michael Somos_, May 07 2017

G. C. Greubel, Table of n, a(n) for n = 1..500
P. Bala, Divisibility sequences from strong divisibility sequences
Wikipedia, Divisibility sequence
Wikipedia, Fibonacci number
Wikipedia, Lucas Sequence
Index entries for linear recurrences with constant coefficients, signature (22,250,-1320,-250,22,1).

Cf. A000045, A215466, A238536, A238600, A238602, A238603.

[(Fibonacci(3*n) + (-1)^n*Fibonacci(5*n) + Fibonacci(7*n))/10: n in [1..30]]; // G. C. Greubel, Aug 07 2018

with(combinat):
seq(1/10*fibonacci(3*n)*fibonacci(5*n)/fibonacci(n), n = 1..20);

Table[(1/10)*(Fibonacci[3*n] + (-1)^n*Fibonacci[5*n] + Fibonacci[7*n]), {n, 0, 50}] (* G. C. Greubel, Aug 07 2018 *)

{a(n) = if(n, fibonacci(3*n) * fibonacci(5*n) / (10 * fibonacci(n)), 0)} /* Michael Somos, May 07 2017 */

a(n) = (1/10)*(Fibonacci(3*n) + (-1)^n*Fibonacci(5*n) + Fibonacci(7*n)).
The sequence can be extended to negative indices using a(-n) = (-1)^(n+1)*a(n).
O.g.f. x*(1 + 22*x - 181*x^2 - 22*x^3 + x^4)/( (1 - 4*x - x^2)*(1 + 11*x - x^2)*(1 - 29*x - x^2) ).
Recurrence equation: a(n) = 22*a(n-1) + 250*a(n-2) - 1320*a(n-3) - 250*a(n-4) + 22*a(n-5) + a(n-6).

A238602 A sixth-order linear divisibility sequence related to the Pell numbers: a(n) := (1/60)Pell(3n)Pell(4n)/Pell(n).

Original entry on oeis.org

1, 238, 45507, 9063516, 1792708805, 355009117386, 70287911575687, 13916722851826872, 2755438412296182921, 545562971271797876390, 108018710075587599558731, 21387159127038457710621972, 4234549485214861760195346253, 838419411023095574089504928386

Offset: 1

Peter Bala, Mar 06 2014

Let P and Q be relatively prime integers. The Lucas sequence U(n) (which depends on P and Q) is an integer sequence that satisfies the recurrence equation a(n) = P*a(n-1) - Q*a(n-2) with the initial conditions U(0) = 0, U(1) = 1. The sequence {U(n)}n>=1 is a strong divisibility sequence, i.e., gcd(U(n),U(m)) = |U(gcd(n,m))|. It follows that {U(n)} is a divisibility sequence, i.e., U(n) divides U(m) whenever n divides m and U(n) <> 0.
It can be shown that if p and q are a pair of relatively prime positive integers, and if U(n) never vanishes, then the sequence {U(p*n)*U(q*n)/U(n)}n>=1 is a linear divisibility sequence of order 2*min(p,q). For a proof and a generalization of this result see the Bala link.
Here we take p = 3 and q = 4 with P = 2 and Q = -1, for which U(n) is the sequence of Pell numbers, A000129, and normalize the sequence {U(3*n)*U(4*n)/U(n)}n>=1 to have the initial term 1.
For other sequences of this type see A238600, A238601 and A238603. See also A238536.

G. C. Greubel, Table of n, a(n) for n = 1..435
P. Bala, Divisibility sequences from strong divisibility sequences
Wikipedia, Divisibility sequence
Wikipedia, Lucas Sequence
Wikipedia, Pell number
Index entries for linear recurrences with constant coefficients, signature (170,5745,-40052,5745,170,-1).

Cf. A000129, A238536, A238600, A238601, A238603.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1 +68*x-698*x^2+68*x^3+x^4)/((1-6*x+x^2)*(1+34*x+x^2)*(1-198*x+x^2)))); // G. C. Greubel, Aug 07 2018

Table[(1/60)*(Fibonacci[2*n, 2] + (-1)^n*Fibonacci[4*n, 2] + Fibonacci[6*n, 2]), {n, 1, 50}] (* G. C. Greubel, Aug 07 2018 *)

x='x+O('x^30); Vec(x*(1+68*x-698*x^2+68*x^3+x^4)/((1-6*x+x^2)*(1 + 34*x+x^2)*(1-198*x+x^2))) \\ G. C. Greubel, Aug 07 2018

a(n) = (1/60)*( Pell(2*n) + (-1)^n*Pell(4*n) + Pell(6*n) ).
The sequence can be extended to negative indices using a(-n) = -a(n).
O.g.f. x*(1 + 68*x - 698*x^2 + 68*x^3 + x^4)/( (1 - 6*x + x^2)*(1 + 34*x + x^2)*(1 - 198*x + x^2) ).
Recurrence equation: a(n) = 170*a(n-1) + 5745*a(n-2) - 40052*a(n-3) + 5745*a(n-4) + 170*a(n-5) - a(n-6).

A238603 A sixth-order linear divisibility sequence related to A000225: a(n) := (1/105)(2^(3n) - 1)(2^(4n) - 1)/(2^n - 1).

Original entry on oeis.org

1, 51, 2847, 170391, 10555655, 664857063, 42215949223, 2691226507047, 171901443816999, 10990938133564455, 703076406514657319, 44985901769992495143, 2878746218051469266983, 184228512166784552153127, 11790264946382521291370535, 754565442462197107544125479

Offset: 1

Peter Bala, Mar 06 2014

Let P and Q be relatively prime integers. The Lucas sequence U(n) (which depends on P and Q) is an integer sequence that satisfies the recurrence equation a(n) = P*a(n-1) - Q*a(n-2) with the initial conditions U(0) = 0, U(1) = 1. The sequence {U(n)}n>=1 is a strong divisibility sequence, i.e., gcd(U(n),U(m)) = |U(gcd(n,m))|. It follows that {U(n)} is a divisibility sequence, i.e., U(n) divides U(m) whenever n divides m and U(n) <> 0.
It can be shown that if p and q are a pair of relatively prime positive integers, and if U(n) never vanishes, then the sequence {U(p*n)*U(q*n)/U(n)}n>=1 is a linear divisibility sequence of order 2*min(p,q). For a proof and a generalization of this result see the Bala link.
Here we take p = 3 and q = 4 with P = 3 and Q = 2, for which U(n) is the sequence A000225 (sometimes called the Mersenne numbers), and normalize the sequence {U(3*n)*U(4*n)/U(n)}n>=1 to have the initial term 1.
For other sequences of this type see A238600, A238601 and A238602. See also A238536.

			G.f. = x + 51*x^2 + 2847*x^3 + 170391*x^4 + 10555655*x^5 + 664857063*x^6 + ... - _Michael Somos_, May 07 2017

G. C. Greubel, Table of n, a(n) for n = 1..500
P. Bala, Divisibility sequences from strong divisibility sequences
Wikipedia, Divisibility sequence
Wikipedia, Fibonacci number
Wikipedia, Lucas Sequence
Index entries for linear recurrences with constant coefficients, signature (119,-4382,59432,-280448,487424,-262144).

Cf. A000225, A238536, A238600, A238601, A238602.

[(1/105)*(64^n + 32^n + 16^n - 4^n - 2^n - 1): n in [1..50]]; // G. C. Greubel, Aug 07 2018

seq(1/105*(2^(3*n)-1)*(2^(4*n)-1)/(2^n-1), n = 1..20);

Table[(1/105)*(64^n + 32^n + 16^n - 4^n - 2^n - 1), {n, 1, 50}] (* G. C. Greubel, Aug 07 2018 *)

{a(n) = if( n, (8^n - 1) * (16^n - 1) / (105 * (2^n - 1)), 0)}; /* Michael Somos, May 07 2017 */

a(n) = (1/105)*(64^n + 32^n + 16^n - 4^n - 2^n - 1).
O.g.f.: x*(4096*x^4 - 4352*x^3 + 1160*x^2 - 68*x + 1 )/( (1-x)*(1-2*x)(1-4*x)*(1-16*x)*(1-32*x)*(1-64*x) ).
The formula for a(n) may be used to define it for all n in Z, and then we have a(n) = -(64)^n * a(-n). - Michael Somos, May 07 2017

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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A238537 A fourth-order linear divisibility sequence related to the Pell numbers.

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A238538 A fourth-order linear divisibility sequence: a(n) = (2^n + 1)*(2^(3*n) - 1)/ ( (2 + 1)*(2^3 - 1) ).

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A238600 A sixth-order linear divisibility sequence related to the Fibonacci numbers: a(n) := (1/6)*Fibonacci(3*n)*Fibonacci(4*n)/Fibonacci(n).

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A238540 A fourth-order linear divisibility sequence: a(n) := (3^n + 1)*(3^(3*n) - 1)/( (3 + 1)*(3^3 - 1)).

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A238541 A fourth-order linear divisibility sequence: a(n) := A(n)/A(1) where A(n) := ( (3^n + 2^n)*(3^(3*n) - 2^(3*n)) ).

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A238539 A fourth-order linear divisibility sequence: a(n) := (1/9)*(2^n + (-1)^n)*(2^(3*n) - (-1)^n).

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

A238538 A fourth-order linear divisibility sequence: a(n) = (2^n + 1)(2^(3n) - 1)/ ( (2 + 1)*(2^3 - 1) ).

A238600 A sixth-order linear divisibility sequence related to the Fibonacci numbers: a(n) := (1/6)Fibonacci(3n)Fibonacci(4n)/Fibonacci(n).

A238540 A fourth-order linear divisibility sequence: a(n) := (3^n + 1)(3^(3n) - 1)/( (3 + 1)*(3^3 - 1)).

A238541 A fourth-order linear divisibility sequence: a(n) := A(n)/A(1) where A(n) := ( (3^n + 2^n)(3^(3n) - 2^(3*n)) ).

A238539 A fourth-order linear divisibility sequence: a(n) := (1/9)(2^n + (-1)^n)(2^(3*n) - (-1)^n).

A238601 A sixth-order linear divisibility sequence related to the Fibonacci numbers: a(n) := (1/10)Fibonacci(3n)Fibonacci(5n)/Fibonacci(n).

A238602 A sixth-order linear divisibility sequence related to the Pell numbers: a(n) := (1/60)Pell(3n)Pell(4n)/Pell(n).

A238603 A sixth-order linear divisibility sequence related to A000225: a(n) := (1/105)(2^(3n) - 1)(2^(4n) - 1)/(2^n - 1).