A238603 A sixth-order linear divisibility sequence related to A000225: a(n) := (1/105)*(2^(3*n) - 1)*(2^(4*n) - 1)/(2^n - 1).

1, 51, 2847, 170391, 10555655, 664857063, 42215949223, 2691226507047, 171901443816999, 10990938133564455, 703076406514657319, 44985901769992495143, 2878746218051469266983, 184228512166784552153127, 11790264946382521291370535, 754565442462197107544125479

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 = 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 )/( (1-x)*(1-2*x)(1-4*x)*(1-16*x)*(1-32*x)*(1-64*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^n-1), 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