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%I #36 Mar 02 2024 14:09:27
%S 0,4,48,448,3840,31744,258048,2080768,16711680,133955584,1072693248,
%T 8585740288,68702699520,549688705024,4397778075648,35183298347008,
%U 281470681743360,2251782633816064,18014329790005248,144114913197948928,1152920405095219200
%N a(n) = 4^n * (2^n - 1).
%C Jordan's totient functions are described more fully in A059379 and A059380; for example, J_1(n) is Euler's totient function and J_2(n) the Moebius transform of squares.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%H Harry J. Smith, <a href="/A059409/b059409.txt">Table of n, a(n) for n = 0..100</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-32).
%F Equals J_n(8) (see A059379).
%F J_n(8) = 8^n - A024023(n) - A000225(n) - A000012(n).
%F a(n) = 4*A016152(n).
%F G.f.: 4*x / ( (8*x-1)*(4*x-1) ). - _R. J. Mathar_, Nov 23 2018
%F Sum_{n>0} 1/a(n) = E - 4/3, where E is the Erdős-Borwein constant (A065442). - _Peter McNair_, Dec 19 2022
%F a(n) = A291779(A008585(n)) = A045991(A000079(n)). - _Mathew Englander_, Feb 08 2024
%e (4,48,448,3840,...) = (8,64,512,4096,...) - (2,12,56,240,...) - (1,3,7,15,...) - (1,1,1,1,...)
%p seq(4^n * (2^n - 1), n=0..20); # _Muniru A Asiru_, Jan 29 2018
%t Table[4^n*(2^n - 1), {n,0,30}] (* _G. C. Greubel_, Jan 29 2018 *)
%t LinearRecurrence[{12,-32},{0,4},20] (* _Harvey P. Dale_, Oct 14 2019 *)
%o (PARI) { for (n = 0, 100, write("b059409.txt", n, " ", 4^n*(2^n - 1)); ) } \\ _Harry J. Smith_, Jun 26 2009
%o (Magma) [4^n*(2^n - 1): n in [0..40]]; // _Vincenzo Librandi_, 26 2011
%o (GAP) List([0..100], n->4^n * (2^n - 1)); # _Muniru A Asiru_, Jan 29 2018
%Y Cf. A059379, A059380, A016152.
%Y Cf. A024023, A000225, A000012.
%Y Cf. A065442.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, _Alford Arnold_, Jan 30 2001
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