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%I #20 Jun 30 2023 14:15:36
%S 1,31,211,1291,7771,46651,279931,1679611,10077691,60466171,362797051,
%T 2176782331,13060694011,78364164091,470184984571,2821109907451,
%U 16926659444731,101559956668411,609359740010491,3656158440062971
%N a(n) = 6^n-5.
%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 Vincenzo Librandi, <a href="/A164784/b164784.txt">Table of n, a(n) for n = 1..1000</a>
%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 Daniel Minoli, 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7, -6).
%F a(n) = 6*a(n-1)+25 with n>1, a(1)=1. - _Vincenzo Librandi_, Oct 29 2009
%F G.f.: x*(1 + 24*x)/(1 - 7*x + 6*x^2). - _Vincenzo Librandi_, Feb 06 2013
%F E.g.f.: 4 + (exp(5*x) - 5)*exp(x). - _Ilya Gutkovskiy_, Jun 11 2016
%t CoefficientList[Series[(1 + 24 x)/(1 - 7 x + 6 x^2), {x, 0, 30}],x] (* _Vincenzo Librandi_, Feb 06 2013 *)
%o (Magma) [6^n-5: n in [1..30]]; // _Vincenzo Librandi_, Feb 06 2013
%K nonn,easy
%O 1,2
%A Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009