A110593 -id:A110593 - cp's OEIS frontend

A110594 a(1) = 4, a(2) = 12, for n>1: a(n) = 3*4^(n-1).

4, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888, 51539607552, 206158430208, 824633720832, 3298534883328, 13194139533312, 52776558133248

Offset: 1

Jonathan Vos Post, Jul 29 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A000302, A007090, A081604, A110591, A110593.

Concatenation([4],List([2..25],n->3*4^(n-1))); # Muniru A Asiru, Oct 21 2018

[4] cat [3*4^(n-1): n in [2..30]]; // Vincenzo Librandi, May 29 2014

seq(coeff(series(4*x*(1-x)/(1-4*x),x,n+1), x, n), n = 1 .. 25); # Muniru A Asiru, Oct 21 2018

CoefficientList[Series[4 (1 - x)/(1 - 4 x), {x, 0, 40}], x] (* Vincenzo Librandi, May 29 2014 *)

x='x+O('x^50); Vec(4*x*(1 - x)/(1 - 4*x)) \\ G. C. Greubel, Sep 01 2017

a(n) = A002001(n), n>1. - R. J. Mathar, Aug 18 2008

G.f.: 4*x*(1 - x)/(1 - 4*x). - Vincenzo Librandi, May 29 2014

Definition corrected by R. J. Mathar, Aug 18 2008

A103457 a(n) = 3^n + 1 - 0^n.

Original entry on oeis.org

1, 4, 10, 28, 82, 244, 730, 2188, 6562, 19684, 59050, 177148, 531442, 1594324, 4782970, 14348908, 43046722, 129140164, 387420490, 1162261468, 3486784402, 10460353204, 31381059610, 94143178828, 282429536482, 847288609444

Offset: 0

Paul Barry, Feb 07 2005

Patrick De Geest, World!Of Numbers
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A011782, A027649, A034472, A094388, A110593.

[1] cat [3^n + 1: n in [1..30]]; // G. C. Greubel, Jun 22 2021

Join[{1},LinearRecurrence[{4,-3},{4,10},30]] (* Harvey P. Dale, Mar 29 2015 *)

my(x='x+O('x^50)); Vec((1-3*x^2)/((1-x)*(1-3*x))) \\ Altug Alkan, Dec 04 2015

[1]+[3^n + 1 for n in (1..30)] # G. C. Greubel, Jun 22 2021

G.f.: (1-3*x^2)/((1-x)*(1-3*x)).
a(n) = Sum_{k=0..n} binomial(n, k)*0^(k(n-k))*3^k.
From R. J. Mathar, Aug 04 2008: (Start)
a(n) = A034472(n), n>0.
a(n) = A094388(n-1), n>1.
a(n+1) - a(n) = A110593(n+1). (End)
a(n) = 3*a(n-1) - 2, with a(1)=4. - Vincenzo Librandi, Dec 29 2010
From J. Conrad, Nov 25 2015: (Start)
For n>0, a(n) = 2 * (A011782(0) + A011782(n) + Sum_{x=1..n-1} Sum_{k=0..x-1}(binomial(x-1,k)*(A011782(k+1) + A011782(n-x+k)))).
Alternatively, for n>0, a(n) = A027649(n) - 2 * Sum_{x=1..n-1}Sum_{k=0..x-1}(binomial(x-1,k)*(A011782(k+1) + A011782(n-x+k))). (End)
E.g.f.: -1 + exp(x) + exp(3*x). - G. C. Greubel, Jun 22 2021

A164907 a(n) = (3*3^n-(-1)^n)/2.

Original entry on oeis.org

1, 5, 13, 41, 121, 365, 1093, 3281, 9841, 29525, 88573, 265721, 797161, 2391485, 7174453, 21523361, 64570081, 193710245, 581130733, 1743392201, 5230176601, 15690529805, 47071589413, 141214768241, 423644304721, 1270932914165

Offset: 0

Klaus Brockhaus, Aug 31 2009

Interleaving of A096053 and A083884 without initial term 1.
Partial sums are (essentially) in A080926.
First differences are (essentially) in A105723.
a(n)+a(n+1) = A008776(n+1) = A099856(n+1) = A110593(n+2).
Binomial transform of A056450. Inverse binomial transform of A164908.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,3).

Equals A046717 without initial term 1 and A080925 without initial term 0. Equals A084182 / 2 from second term onward.

Cf. A096053, A083884, A080926, A105723, A008776, A099856, A110593, A056450, A164908.

Magma
```
[ (3*3^n-(-1)^n)/2: n in [0..25] ];
```

A164907:=n->(3*3^n - (-1)^n)/2; seq(A164907(n), n=0..30); # Wesley Ivan Hurt, Mar 21 2014

Table[(3*3^n - (-1)^n)/2, {n, 0, 30}] (* Wesley Ivan Hurt, Mar 21 2014 *)
LinearRecurrence[{2,3},{1,5},50] (* Harvey P. Dale, Oct 31 2018 *)

a(n) = 2*a(n-1)+3*a(n-2) for n > 1; a(0) = 1, a(1) = 5.
G.f.: (1+3*x)/((1+x)*(1-3*x)).
a(n) = 3*a(n-1)+2*(-1)^n. - Carmine Suriano, Mar 21 2014

A110595 a(1)=5. For n > 1, a(n) = 4*5^(n-1) = A005054(n).

Original entry on oeis.org

5, 20, 100, 500, 2500, 12500, 62500, 312500, 1562500, 7812500, 39062500, 195312500, 976562500, 4882812500, 24414062500, 122070312500, 610351562500, 3051757812500, 15258789062500, 76293945312500, 381469726562500

Offset: 1

Jonathan Vos Post, Jul 29 2005

a(n) is the number of n-digit integers that contain only even digits (A014263). - Bernard Schott, Nov 11 2022

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A000351, A007091, A014263, A081604, A110591, A110592, A110593, A110594.

Join[{5},NestList[5#&,20,20]] (* Harvey P. Dale, Jun 19 2013 *)
Rest[CoefficientList[Series[5 x (1 - x)/(1 - 5 x), {x,0,50}], x]] (* G. C. Greubel, Sep 01 2017 *)

my(x='x+O('x^50)); Vec(5*x*(1-x)/(1-5*x)) \\ G. C. Greubel, Sep 01 2017

O.g.f.: 5*x*(1-x)/(1-5*x). - Better definition from R. J. Mathar, May 13 2008

Sum_{n>=1} 1/a(n) = 21/80. - Bernard Schott, Nov 11 2022

Better definition from R. J. Mathar, May 13 2008

Incorrect comment removed by Michel Marcus, Nov 11 2022

cp's OEIS Frontend

A110594 a(1) = 4, a(2) = 12, for n>1: a(n) = 3*4^(n-1).

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A103457 a(n) = 3^n + 1 - 0^n.

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A164907 a(n) = (3*3^n-(-1)^n)/2.

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A110595 a(1)=5. For n > 1, a(n) = 4*5^(n-1) = A005054(n).

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