A005610 -id:A005610

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A125682

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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).

FORMULA

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Zerinvary Lajos, Jan 31 2007

EXTENSIONS

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

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

STATUS

approved

A125725

Numbers whose base-7 representation is 222....2.

+10
2

0, 2, 16, 114, 800, 5602, 39216, 274514, 1921600, 13451202, 94158416, 659108914, 4613762400, 32296336802, 226074357616, 1582520503314, 11077643523200, 77543504662402, 542804532636816, 3799631728457714, 26597422099204000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Davis Smith, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (8,-7).

FORMULA

a(n) = (7^(n-1) - 1)/3 = 2*A023000(n-1).

a(n) = 7*a(n-1) + 2, with a(1)=0. - Vincenzo Librandi, Sep 30 2010

G.f.: 2*x^2 / ( (1-x)*(1-7*x) ). - R. J. Mathar, Sep 30 2013

From Davis Smith, Apr 04 2019: (Start)

A007310(a(n) + 1) = 7^(n - 1).

A047522(a(n + 1)) = -1*A165759(n). (End)

E.g.f.: (exp(7*x) - 7*exp(x) + 6)/21. - Stefano Spezia, Jan 12 2025

EXAMPLE

base 7.......decimal

0..................0

2..................2

22................16

222..............114

2222.............800

22222...........5602

222222.........39216

2222222.......274514

22222222.....1921600

222222222...13451202

etc...........etc.

MAPLE

seq(2*(7^n-1)/6, n=0..25);

MATHEMATICA

FromDigits[#, 7]&/@Table[PadLeft[{2}, n, 2], {n, 0, 25}] (* Harvey P. Dale, Apr 13 2011 *)

(7^(Range[25]-1) - 1)/3 (* G. C. Greubel, May 23 2019 *)

PROG

(PARI) vector(25, n, (7^(n-1)-1)/3) \\ Davis Smith, Apr 04 2019

(Magma) [0] cat [n:n in [1..15000000]| Set(Intseq(n, 7)) subset [2]]; // Marius A. Burtea, May 06 2019

(Magma) [(7^(n-1)-1)/3: n in [1..25]]; // Marius A. Burtea, May 06 2019

(Sage) [(7^(n-1) -1)/3 for n in (1..25)] # G. C. Greubel, May 23 2019

(GAP) List([1..25], n-> (7^(n-1) -1)/3); # G. C. Greubel, May 23 2019

CROSSREFS

Cf. A007310, A023000, A047522, A165759.

Cf. also A002276, A005610, A020988, A024023, A125831, A125835, A125857 for related or similarly constructed sequences.

KEYWORD

easy,nonn,base

AUTHOR

Zerinvary Lajos, Feb 02 2007

EXTENSIONS

Offset corrected by N. J. A. Sloane, Oct 02 2010

STATUS

approved

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