A164786 a(n) = 8^n-7.

1, 57, 505, 4089, 32761, 262137, 2097145, 16777209, 134217721, 1073741817, 8589934585, 68719476729, 549755813881, 4398046511097, 35184372088825, 281474976710649, 2251799813685241, 18014398509481977, 144115188075855865

Offset: 1

Author

Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009

Keywords

• nonn
• easy

Comments

Minoli defined the sequences and concepts that follow in the 1980 IEEE paper below: - Sequence m(n,t) = (n^t) - (n-1) for t=2 to infinity is called a Mersenne Sequence Rooted on n - If n is prime, this sequence is called a Legitimate Mersenne Sequence - Any j belonging to the sequence m(n,t) is called a Generalized Mersenne Number (n-GMN) - If j belonging to the sequence m (n,t) is prime, it is then called a n-Generalized Mersenne Prime (n-GMP). Note: m (n,t) = n* m (n,t-1) + n^2 - 2*n+1. This sequence related to sequences: A014232 and A014224; A135535 and A059266. These numbers play a role in the context of hyperperfect numbers. For additional references, beyond key ones listed below, see A164783.

References

• Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Daniel Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
• Daniel Minoli, W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
• Index entries for linear recurrences with constant coefficients, signature (9, -8).

Programs

Magma

[8^n-7: n in [1..20]]; // Vincenzo Librandi, Aug 22 2011

Mathematica

8^Range[20]-7 (* or *) LinearRecurrence[{9,-8},{1,57},20] (* Harvey P. Dale, Jan 24 2013 *)

Formula

a(n) = 8*a(n-1)+49, with a(1)=1. - Vincenzo Librandi, Nov 30 2010

G.f.: x*(1+48*x)/(1-9*x+8*x^2). a(n) = 9*a(n-1)-8*a(n-2). - Colin Barker, Jan 28 2012

E.g.f.: 6 + (exp(7*x) - 7)*exp(x). - Ilya Gutkovskiy, Jun 11 2016

Extensions

More terms a(7)-a(19) from Vincenzo Librandi, Oct 29 2009