A164785 a(n) = 5^n - 4.

1, 21, 121, 621, 3121, 15621, 78121, 390621, 1953121, 9765621, 48828121, 244140621, 1220703121, 6103515621, 30517578121, 152587890621, 762939453121, 3814697265621, 19073486328121, 95367431640621, 476837158203121

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

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
• Daniel Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
• Index entries for linear recurrences with constant coefficients, signature (6,-5).

Crossrefs

Cf. A059613.

Programs

Magma

[5^n-4: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013

Mathematica

5^Range[50]-4 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)
LinearRecurrence[{6,-5},{1,21},30] (* or *) NestList[5 # + 16 &, 1, 30] (* Harvey P. Dale, Jun 07 2012 *)
CoefficientList[Series[(1 + 15 x)/(1 - 6 x + 5 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)

Formula

a(n) = 5*a(n-1) + 16 with n > 1, a(1)=1. - Vincenzo Librandi, Nov 30 2010

a(n) = 6*a(n-1) - 5*a(n-2); a(1)=1, a(2)=21. - Harvey P. Dale, Jun 07 2012

G.f.: x*(1 + 15*x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Feb 06 2013

E.g.f.: 3 + (exp(4*x) - 4)*exp(x). - Ilya Gutkovskiy, Jun 11 2016

Extensions

More terms a(9)-a(21) from Vincenzo Librandi, Oct 29 2009