A059409 -id:A059409 - cp's OEIS frontend

A059410 J_n(9) (see A059379).

0, 6, 72, 702, 6480, 58806, 530712, 4780782, 43040160, 387400806, 3486725352, 31380882462, 282429005040, 2541864234006, 22876787671992, 205891117745742, 1853020145805120, 16677181570526406, 150094634909578632, 1350851716510730622, 12157665455570144400, 109418989121052006006

Offset: 0

N. J. A. Sloane, Jan 30 2001

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (12,-27).

Cf. A000244, A016142, A024023, A059379, A059380, A059409.

[9^n-3^n: n in [0..20]]; // Vincenzo Librandi, Jun 03 2011

A059410:=n->9^n-3^n: seq(A059410(n), n=0..30); # Wesley Ivan Hurt, Aug 16 2016

Table[9^n - 3^n, {n, 0, 25}] (* or *) CoefficientList[Series[6 x /((1 - 3 x) (1 - 9 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2014 *)

a(n) = 9^n - 3^n; a(n) = 12*a(n-1) - 27*a(n-2) for n > 1. - Vincenzo Librandi, Jun 03 2011
From Vincenzo Librandi, Oct 04 2014: (Start)
a(n) = 3^n*(3^n-1) = A000244(n)*A024023(n).
G.f.: 6*x/((1-3*x)*(1-9*x)). (End)
a(n) = 6*A016142(n). - R. J. Mathar, Nov 23 2018
E.g.f.: 2*exp(6*x)*sinh(3*x). - Elmo R. Oliveira, Mar 31 2025

A248337 a(n) = 6^n - 4^n.

Original entry on oeis.org

0, 2, 20, 152, 1040, 6752, 42560, 263552, 1614080, 9815552, 59417600, 358602752, 2160005120, 12993585152, 78095728640, 469111242752, 2816814940160, 16909479575552, 101491237191680, 609084862103552, 3655058928435200, 21932552593866752, 131604111656222720, 789659854309425152, 4738099863344906240, 28429162130022858752

Offset: 0

Vincenzo Librandi, Oct 05 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-24).

Cf. sequences of the form k^n - 4^n: -A000302 (k=0), -A024036 (k=1), -A020522 (k=2), -A005061 (k=3), A005060 (k=5), this sequence (k=6), A190542 (k=7), A059409 (k=8), A118004 (k=9), A248338 (k=10), A139742 (k=11), 8*A016159 (k=12).

Cf. A000079, A000302, A000400, A001047, A081199.

Magma
```
[6^n-4^n: n in [0..30]];
```

Table[6^n - 4^n, {n,0,30}]
CoefficientList[Series[(2 x)/((1-4 x)(1-6 x)), {x, 0, 30}], x]
LinearRecurrence[{10,-24},{0,2},30] (* Harvey P. Dale, Aug 18 2024 *)

vector(20,n,6^(n-1)-4^(n-1)) \\ Derek Orr, Oct 05 2014

A248337=BinaryRecurrenceSequence(10,-24,0,2)
[A248337(n) for n in range(31)] # G. C. Greubel, Nov 11 2024

G.f.: 2*x/((1-4*x)*(1-6*x)).
a(n) = 10*a(n-1) - 24*a(n-2).
a(n) = 2^n*(3^n-2^n) =  A000079(n) * A001047(n) = A000400(n) - A000302(n).
a(n) = 2*A081199(n). - Bruno Berselli, Oct 05 2014
E.g.f.: 2*exp(5*x)*sinh(x). - G. C. Greubel, Nov 11 2024

More terms added by G. C. Greubel, Nov 11 2024

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)

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A059410 J_n(9) (see A059379).

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A248337 a(n) = 6^n - 4^n.

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

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