A164784 a(n) = 6^n-5.

1, 31, 211, 1291, 7771, 46651, 279931, 1679611, 10077691, 60466171, 362797051, 2176782331, 13060694011, 78364164091, 470184984571, 2821109907451, 16926659444731, 101559956668411, 609359740010491, 3656158440062971

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 (7, -6).

Programs

Magma

[6^n-5: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013

Mathematica

CoefficientList[Series[(1 + 24 x)/(1 - 7 x + 6 x^2), {x, 0, 30}],x] (* Vincenzo Librandi, Feb 06 2013 *)

Formula

a(n) = 6*a(n-1)+25 with n>1, a(1)=1. - Vincenzo Librandi, Oct 29 2009

G.f.: x*(1 + 24*x)/(1 - 7*x + 6*x^2). - Vincenzo Librandi, Feb 06 2013

E.g.f.: 4 + (exp(5*x) - 5)*exp(x). - Ilya Gutkovskiy, Jun 11 2016