

A238603


A sixthorder linear divisibility sequence related to A000225: a(n) := (1/105)*(2^(3*n)  1)*(2^(4*n)  1)/(2^n  1).


4



1, 51, 2847, 170391, 10555655, 664857063, 42215949223, 2691226507047, 171901443816999, 10990938133564455, 703076406514657319, 44985901769992495143, 2878746218051469266983, 184228512166784552153127, 11790264946382521291370535, 754565442462197107544125479
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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(n1)  Q*a(n2) 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.


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 (119,4382,59432,280448,487424,262144).


FORMULA

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 )/( (1x)*(12*x)(14*x)*(116*x)*(132*x)*(164*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


EXAMPLE

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


MAPLE

seq(1/105*(2^(3*n)1)*(2^(4*n)1)/(2^n1), n = 1..20);


MATHEMATICA

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


PROG

(PARI) {a(n) = if( n, (8^n  1) * (16^n  1) / (105 * (2^n  1)), 0)}; /* Michael Somos, May 07 2017 */
(Magma) [(1/105)*(64^n + 32^n + 16^n  4^n  2^n  1): n in [1..50]]; // G. C. Greubel, Aug 07 2018


CROSSREFS

Cf. A000225, A238536, A238600, A238601, A238602.
Sequence in context: A097836 A267786 A267733 * A128917 A172742 A172821
Adjacent sequences: A238600 A238601 A238602 * A238604 A238605 A238606


KEYWORD

nonn,easy


AUTHOR

Peter Bala, Mar 06 2014


STATUS

approved



