A273623 -id:A273623 - cp's OEIS frontend

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

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

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

cp's OEIS Frontend

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

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

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

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cp's OEIS Frontend

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

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Author

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Programs

Magma

Maple

Mathematica

PARI

Formula

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

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Author

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Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

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

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

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