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%I #35 Sep 08 2022 08:45:47
%S 1,21,121,621,3121,15621,78121,390621,1953121,9765621,48828121,
%T 244140621,1220703121,6103515621,30517578121,152587890621,
%U 762939453121,3814697265621,19073486328121,95367431640621,476837158203121
%N a(n) = 5^n - 4.
%C 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.
%D Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)
%H Harvey P. Dale, <a href="/A164785/b164785.txt">Table of n, a(n) for n = 1..1000</a>
%H Daniel Minoli and W. Nakamine, <a href="http://dx.doi.org/10.1109/ICASSP.1980.1170906">Mersenne Numbers Rooted On 3 For Number Theoretic Transforms</a>, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
%H Daniel Minoli and Robert Bear, <a href="http://www.pme-math.org/journal/issues/PMEJ.Vol.6.No.3.pdf">Hyperperfect Numbers</a>, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-5).
%F a(n) = 5*a(n-1) + 16 with n > 1, a(1)=1. - _Vincenzo Librandi_, Nov 30 2010
%F a(n) = 6*a(n-1) - 5*a(n-2); a(1)=1, a(2)=21. - _Harvey P. Dale_, Jun 07 2012
%F G.f.: x*(1 + 15*x)/(1 - 6*x + 5*x^2). - _Vincenzo Librandi_, Feb 06 2013
%F E.g.f.: 3 + (exp(4*x) - 4)*exp(x). - _Ilya Gutkovskiy_, Jun 11 2016
%t 5^Range[50]-4 (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2011 *)
%t LinearRecurrence[{6,-5},{1,21},30] (* or *) NestList[5 # + 16 &, 1, 30] (* _Harvey P. Dale_, Jun 07 2012 *)
%t CoefficientList[Series[(1 + 15 x)/(1 - 6 x + 5 x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Feb 06 2013 *)
%o (Magma) [5^n-4: n in [1..30]]; // _Vincenzo Librandi_, Feb 06 2013
%Y Cf. A059613.
%K nonn,easy
%O 1,2
%A Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009
%E More terms a(9)-a(21) from _Vincenzo Librandi_, Oct 29 2009
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