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A273623 a(n) = Fibonacci(3*n) - (2 + (-1)^n)*Fibonacci(n). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

LINKS

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).

FORMULA

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

MAPLE

#A273623

with(combinat):

seq(fibonacci(3*n) - (2 + (-1)^n)*fibonacci(n), n = 1..25);

MATHEMATICA

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 *)

PROG

(MAGMA) [Fibonacci(3*n)-(2+(-1)^n)*Fibonacci(n): n in [1..25]]; // Vincenzo Librandi, Jun 02 2016

(PARI) a(n)=fibonacci(3*n) - (2 + (-1)^n)*fibonacci(n) \\ Charles R Greathouse IV, Jun 08 2016

CROSSREFS

Cf. A000032, A000045, A127595, A273622, A273624, A273625.

Sequence in context: A089574 A077207 A319790 * A001589 A271903 A177467

Adjacent sequences:  A273620 A273621 A273622 * A273624 A273625 A273626

KEYWORD

nonn,easy

AUTHOR

Peter Bala, May 29 2016

STATUS

approved

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Last modified October 19 21:04 EDT 2019. Contains 328225 sequences. (Running on oeis4.)