A147816 -id:A147816 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

Second column of triangle A075499.
Cf. A019677, A147538, A147816.
Cf. A065442, A092237.

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

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

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

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

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

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

A369405 Context-free language 1^n.0^(2n).

Original entry on oeis.org

100, 110000, 111000000, 111100000000, 111110000000000, 111111000000000000, 111111100000000000000, 111111110000000000000000, 111111111000000000000000000, 111111111100000000000000000000, 111111111110000000000000000000000, 111111111111000000000000000000000000

Offset: 1

Vyom Narsana, Jan 22 2024

This sequence represents the context-free language 1^n.0^(2n) which can be accepted by a pushdown automaton. It finds applications in the study of formal languages and automata theory in theoretical computer science.

Paolo Xausa, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (1100,-100000).

Cf. A007088, A059409, A138119, A147816.

a:= n-> convert(4^n*(2^n-1), binary):
seq(a(n), n=1..15);  # Alois P. Heinz, Feb 04 2024

Array[(10^#-1)*10^(2*#)/9 &, 20] (* or *)
LinearRecurrence[{1100, -100000}, {100, 110000}, 20] (* Paolo Xausa, Feb 27 2024 *)

def A369405(n): return (10**n-1)//9*10**(n<<1) # Chai Wah Wu, Feb 11 2024

From Robert Israel, Jan 22 2024: (Start)
a(n) = (10^n-1)*10^(2*n)/9.
G.f.: 100*x/(100000*x^2 - 1100*x + 1). (End)
From Alois P. Heinz, Feb 04 2024: (Start)
a(n) = A007088(A059409(n)).
a(n) = 10 * A138119(n).
a(n) = 100 * A147816(n). (End)

A275944 Gaussian binomial coefficient [n,3] for q = 10.

Original entry on oeis.org

1, 1111, 1122211, 1123333211, 1123445443211, 1123456666543211, 1123457788877543211, 1123457901110987543211, 1123457912334332087543211, 1123457913456666543087543211, 1123457913568899988653087543211, 1123457913580123333209753087543211, 1123457913581245667665420753087543211

Offset: 3

Ilya Gutkovskiy, Aug 13 2016

More generally, the ordinary generation function for the Gaussian binomial coefficients [n,k]q is x^k/Product{m=0..k} (1 - q^m*x).
Convolution of A002275 and A147816 (considering offset: 0, 0, 1, 1100, 1110000, ...).
The first seven members are palindromes.

Index entries related to Gaussian binomial coefficients
Index entries for linear recurrences with constant coefficients, signature (1111,-112110,1111000,-1000000)

Cf. A002275, A022174, A109242, A147816.

Table[((10^n - 100) (10^n - 10) (10^n - 1))/890109000, {n, 0, 15}]
Table[QBinomial[n, 3, 10], {n, 3, 15}]

O.g.f.: x^3/((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)).
E.g.f.: (-1000 + 1110*exp(9*x) - 111*exp(99*x) + exp(999*x))*exp(x)/890109000.
a(n) = 1111*a(n-1) - 112110*a(n-2) + 1111000*a(n-3) - 1000000*a(n-4).
a(n) = ((10^n - 100)*(10^n - 10)*(10^n - 1))/890109000.
a(n) = Product_{i=0..2} (1 - 10^(n-i))/(1 - 10^(i+1)).

cp's OEIS Frontend

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

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A369405 Context-free language 1^n.0^(2n).

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A275944 Gaussian binomial coefficient [n,3] for q = 10.

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