OFFSET
1,2
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).
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
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 *)
PROG
(Magma) [5^n-4: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009
EXTENSIONS
More terms a(9)-a(21) from Vincenzo Librandi, Oct 29 2009
STATUS
approved