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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). 4
1, 44, 1037, 32472, 915305, 26874892, 776952553, 22595381424, 655633561309, 19040507781020, 552780012054689, 16050219184005336, 466002944275859873, 13530204273746536948, 392841165312292809085, 11405932444267712654688, 331164788382150547106857, 9615185834308570310716196 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

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).

FORMULA

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).

EXAMPLE

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

MAPLE

with(combinat):

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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

Sequence in context: A295251 A295650 A004423 * A172978 A114170 A130645

Adjacent sequences:  A238598 A238599 A238600 * A238602 A238603 A238604

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Mar 06 2014

STATUS

approved

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Last modified May 17 22:15 EDT 2021. Contains 343992 sequences. (Running on oeis4.)