A215466 -id:A215466 - 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

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).

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).

A215465 a(n) = (Lucas(4n) - Lucas(2n))/4.

Original entry on oeis.org

0, 1, 10, 76, 540, 3751, 25840, 177451, 1217160, 8344876, 57202750, 392089501, 2687463360, 18420257701, 126254611990, 865362736876, 5931286406640, 40653646980451, 278644255208560, 1909856172864751, 13090349042248500

Offset: 0

R. J. Mathar, Aug 11 2012

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 A085695 (case k = -3). - Peter Bala, Jan 17 2014

In general, 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. See A273622 for the case r = 3, s = 1. - Peter Bala, May 27 2016

			a(3) = 76 because the 12th (4 * 3rd) Lucas number is 22, the 6th (2 * 3rd) Lucas number is 18, and (322 - 18)/4 = 304/4 = 76.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Bala, Lucas sequences and 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.
Hugh Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277
Index entries for linear recurrences with constant coefficients, signature (10,-23,10,-1).

Cf. A085695, A215466, A273622.

[(Lucas(4*n) - Lucas(2*n))/4: n in [0..20]]; // Vincenzo Librandi, Dec 23 2012

A215465 := proc(n)
    (A000032(4*n)-A000032(2*n))/4 ;
end proc:

Table[(LucasL[4n] - LucasL[2n])/4, {n, 0, 19}] (* Alonso del Arte, Aug 11 2012 *)
CoefficientList[Series[-x*(x-1)*(1+x)/((x^2 - 7*x + 1)* (x^2 - 3*x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 23 2012 *)
LinearRecurrence[{10,-23,10,-1},{0,1,10,76},50] (* G. C. Greubel, Jun 02 2016 *)

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

4*a(n) = A000032(4*n) - A000032(2*n).
a(n) = A056854(n)/4 - A005248(n)/4.
G.f.: -x*(x-1)*(1+x) / ( (x^2-7*x+1)*(x^2-3*x+1) ).
a(n) = 10*a(n-1) - 23*a(n-2) + 10*a(n-3) - a(n-4). - G. C. Greubel, Jun 02 2016
a(n) = 2^(-2-n)*((7-3*sqrt(5))^n-(3-sqrt(5))^n-(3+sqrt(5))^n+(7+3*sqrt(5))^n). - Colin Barker, Jun 02 2016
a(n) = a(-n) for all n in Z. - Michael Somos, Dec 29 2022

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).

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)).

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

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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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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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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A215465 a(n) = (Lucas(4n) - Lucas(2n))/4.

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

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).

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)) ).

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

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

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