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A123672 a(1) = 1; for n > 1, a(n) = (2^n-1)*a(n-1) + (-1)^n.

%I A123672 #27 Sep 08 2022 08:45:28
%S A123672 1,4,27,406,12585,792856,100692711,25676641306,13120763707365,
%T A123672 13422541272634396,27475941985082608611,112513982428913282262046,
%U A123672 921602030075228695008418785,15098606058722471710322924954656,494736024726159230532151281989213151
%N A123672 a(1) = 1; for n > 1, a(n) = (2^n-1)*a(n-1) + (-1)^n.
%C A123672 This sequence allows us to prove that the constant C defined in A048651 is irrational. Indeed, for any n > 1 we get |(C+1)*A005329(n) - a(n)| < 1/2^n.
%H A123672 G. C. Greubel, <a href="/A123672/b123672.txt">Table of n, a(n) for n = 1..80</a>
%H A123672 J. Lynch et al., <a href="http://www.jstor.org/stable/2321231">Problem 6233: Irrationality of an infinite product</a>, Amer. Math. Monthly 87 (1980) 408-409
%F A123672 a(n) ~ c * 2^(n*(n+1)/2), where c = 0.372186658950350942813441530084543367... . - _Vaclav Kotesovec_, Oct 10 2016
%t A123672 RecurrenceTable[{a[n] == (2^n - 1) * a[n - 1] + (-1)^n, a[1] == 1}, a, {n, 1, 15}] (* _Vaclav Kotesovec_, Oct 10 2016 *)
%o A123672 (PARI) a(n)=if(n<2,1,(2^n-1)*a(n-1)+(-1)^n)
%o A123672 (Magma) [n eq 1 select 1 else (2^n-1)*Self(n-1)+(-1)^n: n in [1..15]]; // _Vincenzo Librandi_, Oct 18 2017
%Y A123672 Cf. A005329, A048651, A277357.
%K A123672 nonn
%O A123672 1,2
%A A123672 _Benoit Cloitre_, Nov 16 2006