A059409 a(n) = 4^n * (2^n - 1).

0, 4, 48, 448, 3840, 31744, 258048, 2080768, 16711680, 133955584, 1072693248, 8585740288, 68702699520, 549688705024, 4397778075648, 35183298347008, 281470681743360, 2251782633816064, 18014329790005248, 144114913197948928, 1152920405095219200, 9223367638808264704

Offset

0,2

Comments

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.

References

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Links

Harry J. Smith, Table of n, a(n) for n = 0..100

Index entries for linear recurrences with constant coefficients, signature (12,-32).

Formula

Equals J_n(8) (see A059379).

J_n(8) = 8^n - A024023(n) - A000225(n) - A000012(n).

a(n) = 4*A016152(n).

G.f.: 4*x/((8*x-1)*(4*x-1)). - R. J. Mathar, Nov 23 2018

Sum_{n>0} 1/a(n) = E - 4/3, where E is the Erdős-Borwein constant (A065442). - Peter McNair, Dec 19 2022

a(n) = A291779(A008585(n)) = A045991(A000079(n)). - Mathew Englander, Feb 08 2024

From Elmo R. Oliveira, Nov 20 2025: (Start)

E.g.f.: 2*exp(6*x)*sinh(2*x).

a(n) = 2*A147538(n).

a(n) = 12*a(n-1) - 32*a(n-2). (End)

Example

(4,48,448,3840,...) = (8,64,512,4096,...) - (2,12,56,240,...) - (1,3,7,15,...) - (1,1,1,1,...)

Maple

seq(4^n * (2^n - 1), n=0..20); # Muniru A Asiru, Jan 29 2018

Mathematica

Table[4^n*(2^n - 1), {n, 0, 30}] (* G. C. Greubel, Jan 29 2018 *)
LinearRecurrence[{12, -32}, {0, 4}, 20] (* Harvey P. Dale, Oct 14 2019 *)

Prog

(PARI) a(n) = { 4^n*(2^n - 1) } \\ Harry J. Smith, Jun 26 2009
(Magma) [4^n*(2^n - 1): n in [0..40]]; // Vincenzo Librandi, Apr 26 2011
(GAP) List([0..100], n->4^n * (2^n - 1)); # Muniru A Asiru, Jan 29 2018

Crossrefs

Cf. A016152, A059379, A059380.

Cf. A000012, A000225, A024023.

Cf. A000079, A008585, A045991, A065442, A147538, A291779.

Sequence in context: A269180 A228701 A111903 * A357663 A297816 A297987

Adjacent sequences: A059406 A059407 A059408 * A059410 A059411 A059412

Keyword

nonn,easy

Author

N. J. A. Sloane, Alford Arnold, Jan 30 2001

Status

approved