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

0, 1, 12, 112, 960, 7936, 64512, 520192, 4177920, 33488896, 268173312, 2146435072, 17175674880, 137422176256, 1099444518912, 8795824586752, 70367670435840, 562945658454016, 4503582447501312, 36028728299487232

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

Comments

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

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

a(n) is a maximum number of intercalates in a Latin square of order 2^n (see A092237). - Eduard I. Vatutin, Apr 30 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..140
• Eduard I. Vatutin, Example of Latin squares of order 2^n with maximum number of intercalates.
• Index entries for linear recurrences with constant coefficients, signature (12,-32).

Crossrefs

Second column of triangle A075499.

Cf. A019677, A147538, A147816.

Cf. A065442, A092237.

Programs

Magma

[4^(n-1)*(2^n-1): n in [0..40]]; // Vincenzo Librandi, Apr 26 2011

Mathematica

Table[4^(n - 1) (2^n - 1), {n, 0, 19}] (* Michael De Vlieger, Nov 30 2015 *)

PARI

a(n)=4^(n-1)*(2^n-1) \\ Charles R Greathouse IV, Oct 07 2015

PARI

my(x='x+O('x^30)); concat(0, Vec(x/((1-4*x)*(1-8*x)))) \\ Altug Alkan, Dec 04 2015

Sage

[lucas_number1(n,12,32) for n in range(0, 20)] # Zerinvary Lajos, Apr 27 2009

Formula

From Barry E. Williams, Jan 17 2000: (Start)

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

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

a(n) = (4^(n-1))*Stirling2(n+1, 2), n>=0, with Stirling2(n, m)=A008277(n, m).

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

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).

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

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

a(n) + A160873(n) + A006096(n) = A006096(n+2), for n > 2. - Álvar Ibeas, Nov 29 2015

Sum_{n>0} 1/a(n) = 4*E - 16/3, where E is the Erdős-Borwein constant (A065442). - Peter McNair, Dec 19 2022