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%I #24 Sep 08 2022 08:45:47
%S 1,57,505,4089,32761,262137,2097145,16777209,134217721,1073741817,
%T 8589934585,68719476729,549755813881,4398046511097,35184372088825,
%U 281474976710649,2251799813685241,18014398509481977,144115188075855865
%N a(n) = 8^n-7.
%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="/A164786/b164786.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 (9, -8).
%F a(n) = 8*a(n-1)+49, with a(1)=1. - _Vincenzo Librandi_, Nov 30 2010
%F G.f.: x*(1+48*x)/(1-9*x+8*x^2). a(n) = 9*a(n-1)-8*a(n-2). - _Colin Barker_, Jan 28 2012
%F E.g.f.: 6 + (exp(7*x) - 7)*exp(x). - _Ilya Gutkovskiy_, Jun 11 2016
%t 8^Range[20]-7 (* or *) LinearRecurrence[{9,-8},{1,57},20] (* _Harvey P. Dale_, Jan 24 2013 *)
%o (Magma) [8^n-7: n in [1..20]]; // _Vincenzo Librandi_, Aug 22 2011
%K nonn,easy
%O 1,2
%A Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009
%E More terms a(7)-a(19) from _Vincenzo Librandi_, Oct 29 2009
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