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A016152 a(n) = 4^(n-1)*(2^n-1). 8

%I

%S 0,1,12,112,960,7936,64512,520192,4177920,33488896,268173312,

%T 2146435072,17175674880,137422176256,1099444518912,8795824586752,

%U 70367670435840,562945658454016,4503582447501312,36028728299487232

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

%C Numbers whose binary representation is the concatenation of n digits 1 and 2(n-1) digits 0, for n>0. (See A147816.) - _Omar E. Pol_, Nov 13 2008

%C a(n) is the number of lattices L in Z^n such that the quotient group Z^n / L is C_8. - _Álvar Ibeas_, Nov 29 2015

%H Vincenzo Librandi, <a href="/A016152/b016152.txt">Table of n, a(n) for n = 0..140</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12, -32).

%F From _Barry E. Williams_, Jan 17 2000: (Start)

%F a(n) = ((8^(n+1)) - 4^(n+1))/4.

%F a(n) = 12a(n-1) - 32a(n-2), n>0; a(0)=1. (End)

%F a(n) = (4^(n-1))*stirling2(n+1, 2), n>=0, with stirling2(n, m)=A008277(n, m).

%F a(n) = -4^(n-1) + 2*8^(n-1).

%F E.g.f. for a(n+1), n>=0: d^2/dx^2((((exp(4*x)-1)/4)^2)/2!) = -exp(4*x) + 2*exp(8*x).

%F G.f.: x/((1-4*x)*(1-8*x)).

%F ((6+sqrt4)^n - (6-sqrt4)^n)/4 in Fibonacci form. Offset 1. a(3)=112. - Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008

%F a(n) + A160873(n) + A006096(n) = A006096(n+2), for n > 2. - _Álvar Ibeas_, Nov 29 2015

%t Table[4^(n - 1) (2^n - 1), {n, 0, 19}] (* _Michael De Vlieger_, Nov 30 2015 *)

%o (Sage) [lucas_number1(n,12,32) for n in xrange(0, 20)] # _Zerinvary Lajos_, Apr 27 2009

%o (MAGMA) [4^(n-1)*(2^n-1): n in [0..40]]; // _Vincenzo Librandi_, Apr 26 2011

%o (PARI) a(n)=4^(n-1)*(2^n-1) \\ _Charles R Greathouse IV_, Oct 07 2015

%o (PARI) x='x+O('x^100); concat(0, Vec(x/((1-4*x)*(1-8*x)))) \\ _Altug Alkan_, Dec 04 2015

%Y Second column of triangle A075499.

%Y Cf. A019677, A147538, A147816.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

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Last modified March 21 01:18 EDT 2019. Contains 321356 sequences. (Running on oeis4.)