A024062 -id:A024062 - cp's OEIS frontend

A164559 a(n) = 6^n/3 - 1.

1, 11, 71, 431, 2591, 15551, 93311, 559871, 3359231, 20155391, 120932351, 725594111, 4353564671, 26121388031, 156728328191, 940369969151, 5642219814911, 33853318889471, 203119913336831, 1218719480020991, 7312316880125951, 43873901280755711, 263243407684534271

Offset: 1

Klaus Brockhaus, Aug 16 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A000400 (powers of 6), A024062 (6^n-1), A164560.

Magma
```
[ 6^n/3-1: n in [1..20] ];
```

a(n) = 6*a(n-1)+5 for n > 2; a(1) = 1, a(2) = 11.

G.f.: x*(1+4*x)/((1-x)*(1-6*x)).

A125682 a(n) = 3*(6^n - 1)/5.

Original entry on oeis.org

3, 21, 129, 777, 4665, 27993, 167961, 1007769, 6046617, 36279705, 217678233, 1306069401, 7836416409, 47018498457, 282110990745, 1692665944473, 10155995666841, 60935974001049, 365615844006297, 2193695064037785, 13162170384226713, 78973022305360281, 473838133832161689

Offset: 1

Zerinvary Lajos, Jan 31 2007

The base-6 numbers 3_6, 33_6, 333_6, 3333_6, 33333_6, 333333_6, ... converted to base 10.

Also the total number of holes in a certain triangle fractal (start with 6 triangles, 3 holes) after n iterations. See illustration in Ngaokrajang link. - Jens Ahlström, Aug 29 2023

			Base 6        Base 10
3 ............. 3 = 3*6^0
33 ........... 21 = 3*6^1 + 3*6^0
333 ......... 129 = 3*6^2 + 3*6^1 + 3*6^0
3333 ........ 777 = 3*6^3 + 3*6^2 + 3*6^1 + 3*6^0, etc.

Bruno Berselli, Table of n, a(n) for n = 1..1000
Kival Ngaokrajang, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A000400, A003464, A005610, A024062, A105281, A125687, A146884.

[(6^n-1)*3/5: n in [1..22]]; // Bruno Berselli, Apr 18 2012

Maple
```
seq((6^n-1)*3/5, n=1..27);
```

a[n_]:=(6^n-1)*3/5; Table[a[n],{n,1,22}] (* Robert P. P. McKone, Aug 29 2023 *)

G.f.: 3*x/((1-x)*(1-6*x)). - Bruno Berselli, Apr 18 2012
a(n) = 7*a(n-1) - 6*a(n-2). - Wesley Ivan Hurt, Dec 25 2021
From Elmo R. Oliveira, Mar 29 2025: (Start)
E.g.f.: 3*exp(x)*(exp(5*x) - 1)/5.
a(n) = 3*A003464(n). (End)

Edited by N. J. A. Sloane, Feb 02 2007

Definition rewritten (with Lajos formula) from Bruno Berselli, Apr 18 2012

A164560 Partial sums of A164532.

Original entry on oeis.org

1, 5, 11, 35, 71, 215, 431, 1295, 2591, 7775, 15551, 46655, 93311, 279935, 559871, 1679615, 3359231, 10077695, 20155391, 60466175, 120932351, 362797055, 725594111, 2176782335, 4353564671, 13060694015, 26121388031, 78364164095

Offset: 1

Klaus Brockhaus, Aug 16 2009

Interleaving of A164559 and A024062 without initial term 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..900
Index entries for linear recurrences with constant coefficients, signature (1,6,-6).

Cf. A164532, A164123 (partial sums of A162436), A164559 (6^n/3-1), A024062 (6^n-1), A026549.

T:=[ n le 2 select 3*n-2 else 6*Self(n-2): n in [1..28] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];

a(n) = 6*a(n-2)+5 for n > 2; a(1) = 1, a(2) = 5.
a(n) = (3-(-1)^n)*6^(1/4*(2*n-1+(-1)^n))/2-1.
G.f.: x*(1+4*x)/((1-x)*(1-6*x^2)).
a(n) = A026549(n) - 1.

cp's OEIS Frontend

A164559 a(n) = 6^n/3 - 1.

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A125682 a(n) = 3*(6^n - 1)/5.

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A164560 Partial sums of A164532.

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