A146529 -id:A146529 - cp's OEIS frontend

A153973 a(n) = 3a(n-1) - 2a(n-2), with a(1) = 9, a(2) = 12.

9, 12, 18, 30, 54, 102, 198, 390, 774, 1542, 3078, 6150, 12294, 24582, 49158, 98310, 196614, 393222, 786438, 1572870, 3145734, 6291462, 12582918, 25165830, 50331654, 100663302, 201326598, 402653190, 805306374, 1610612742, 3221225478

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A146529, A153972

I:=[9,12]; [n le 2 select I[n] else 3*Self(n-1)-2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Sep 01 2016

a=9;lst={a};Do[a=(a-2)*2-2;AppendTo[lst,a],{n,6!}];lst
NestList[2#-6&,9,30] (* or *) LinearRecurrence[{3,-2},{9,12},31]
Table[ (3/2)*(4 + 2^n), {n, 1, 25}] (* G. C. Greubel, Sep 01 2016 *)

a(n) = 3*a(n-1) - 2*a(n-2), with a(1) = 9, a(2) = 12. - Harvey P. Dale, May 09 2012
From G. C. Greubel, Sep 01 2016: (Start)
a(n) = (3/2)*(4 + 2^n).
G.f.: 3*x*(3 - 5*x)/((1 - x)*(1 - 2*x)).
E.g.f.: (3/2)*(-5 + 4*exp(x) + exp(2*x)). (End)

Definition adapted to offset by Georg Fischer, Jun 18 2021

A194455 a(n) = 2^n + 3n + 1.

Original entry on oeis.org

2, 6, 11, 18, 29, 48, 83, 150, 281, 540, 1055, 2082, 4133, 8232, 16427, 32814, 65585, 131124, 262199, 524346, 1048637, 2097216, 4194371, 8388678, 16777289, 33554508, 67108943, 134217810, 268435541, 536871000, 1073741915, 2147483742, 4294967393, 8589934692, 17179869287

Offset: 0

Bruno Berselli, Sep 01 2011

Inverse binomial transform of this sequence: 2,4,1,1 (1 continued).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000051, A005126, A176691; A120845.
Cf. A062709 (first differences), A000079 (second and successive differences).
Cf. A146529 (differences between alternate terms, for n>2).
Cf. A086653, A115067.

Magma
```
[2^n+3*n+1: n in [0..31]];
```

Table[2^n + 3 n + 1, {n, 0, 40}] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{4,-5,2},{2,6,11},40] (* Harvey P. Dale, Oct 01 2014 *)

PARI
```
for(n=0, 31, print1(2^n+3*n+1", "));
```

G.f.: (2 - 2*x - 3*x^2)/((1 - 2*x)*(1 - x)^2).
a(n) = A086653(n) - 1  for n > 0.
Sum_{i=0..n} a(i) = A115067(n+1) + 2^(n+1).
a(n) = 3*a(n-1) - 2*a(n-2) - 3  for n > 1.
a(n)^2 = 2^(n+1)*(a(n-1) + 3) + (3*n + 1)^2  for n > 2.
E.g.f.: exp(x)*(1 + exp(x) + 3*x). - Stefano Spezia, May 06 2023

cp's OEIS Frontend

A153973 a(n) = 3a(n-1) - 2a(n-2), with a(1) = 9, a(2) = 12.

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A194455 a(n) = 2^n + 3n + 1.

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cp's OEIS Frontend

A153973 a(n) = 3*a(n-1) - 2*a(n-2), with a(1) = 9, a(2) = 12.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A194455 a(n) = 2^n + 3n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A153973 a(n) = 3a(n-1) - 2a(n-2), with a(1) = 9, a(2) = 12.