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A023001 a(n) = (8^n - 1)/7. 73
0, 1, 9, 73, 585, 4681, 37449, 299593, 2396745, 19173961, 153391689, 1227133513, 9817068105, 78536544841, 628292358729, 5026338869833, 40210710958665, 321685687669321, 2573485501354569, 20587884010836553, 164703072086692425 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Gives the (zero-based) positions of odd terms in A007556 (numbers n such that A007556(a(n)) mod 2 = 1). - Farideh Firoozbakht, Jun 13 2003

{1, 9, 73, 585, 4681, ...} is the binomial transform of A003950. - Philippe Deléham, Jul 22 2005

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=8, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n) = det(A). - Milan Janjic, Feb 21 2010

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=9, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n-1) = (-1)^n*charpoly(A,1). - Milan Janjic, Feb 21 2010

This is the sequence A(0,1;7,8;2) = A(0,1;8,0;1) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

a(n) is the total number of squares the carpetmaker has removed after the n-th step of a Sierpiński carpet production. - Ivan N. Ianakiev, Oct 22 2013

For n >= 1, a(n) is the total number of holes in a box fractal (start with 8 boxes, 1 hole) after n iterations. See illustration in link. - Kival Ngaokrajang, Jan 27 2015

From Bernard Schott, May 01 2017: (Start)

Except for 0, 1 and 73, all the terms are composite because a(n) = ((2^n - 1) * (4^n + 2^n + 1))/7.

For n >= 3, all terms are Brazilian repunits numbers in base 8, and so belong to A125134.

a(3) = 73 is the only Brazilian prime in base 8, and so it belongs to A085104 and A285017. (End)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.

Roger B. Eggleton, Maximal Midpoint-Free Subsets of Integers, International Journal of Combinatorics Volume 2015, Article ID 216475, 14 pages.

Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. - Wolfdieter Lang, Oct 18 2010

Kival Ngaokrajang, Illustration of initial terms

Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.

Eric Weisstein's World of Mathematics, Repunit.

Index entries for linear recurrences with constant coefficients, signature (9,-8).

FORMULA

Also sum of cubes of divisors of 2^(n-1): a(n) = A001158(A000079(n-1)). - Labos Elemer, Apr 10 2003 and Farideh Firoozbakht, Jun 13 2003

a(n) = A033138(3n-2). - Alexandre Wajnberg, May 31 2005

From Philippe Deléham, Oct 12 2006: (Start)

a(0) = 0, a(n) = 8*a(n-1) + 1 for n>0.

G.f.: x/((1-8x)*(1-x)). (End)

From Wolfdieter Lang, Oct 18 2010: (Start)

a(n) = 7*a(n-1) + 8*a(n-2) + 2, a(0)=0, a(1)=1.

a(n) = 8*a(n-1) + a(n-2) - 8*a(n-3) = 9*a(n-1) - 8*a(n-2), a(0)=0, a(1)=1, a(2)=9. Observation by Gary Detlefs. See the W. Lang comment and link. (End)

a(n) = Sum_{k=0..n-1} 8^k. - Doug Bell, May 26 2017

EXAMPLE

From Zerinvary Lajos, Jan 14 2007: (Start)

Octal.............decimal

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

1....................1

11...................9

111.................73

1111...............585

11111.............4681

111111...........37449

1111111.........299593

11111111.......2396745

111111111.....19173961

1111111111...153391689

etc. ...............etc. (End)

a(4) = (8^4 - 1)/7 = 585 = 1111_8 = {(2^4 - 1) * (4^4 + 2^4 + 1) /7 = 15 * 273/7 = 15 * 39. - Bernard Schott, May 01 2017

MAPLE

a:=n->sum(8^(n-j), j=1..n): seq(a(n), n=0..20); # Zerinvary Lajos, Jan 04 2007

MATHEMATICA

Table[(8^n-1)/7, {n, 0, m}]

LinearRecurrence[{9, -8}, {0, 1}, 30] (* Harvey P. Dale, Feb 12 2015 *)

PROG

(Sage) [lucas_number1(n, 9, 8) for n in xrange(0, 21)] # Zerinvary Lajos, Apr 23 2009

(Sage) [gaussian_binomial(n, 1, 8) for n in xrange(0, 21)] # Zerinvary Lajos, May 28 2009

(MAGMA) [(8^n-1)/7: n in [0..20]]; // Vincenzo Librandi, Sep 17 2011

(Maxima) A023001(n):=floor((8^n-1)/7)$

makelist(A023001(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */

(PARI) a(n)=(8^n-1)/7 \\ Charles R Greathouse IV, Mar 22 2016

(GAP)

A023001:=List([0..10^2], n->(8^n-1)/7); # Muniru A Asiru, Oct 03 2017

CROSSREFS

Cf. A007556, A003950, A001158, A033138.

Cf. A125134, A085104, A285017, A220571, A053696.

Sequence in context: A126641 A081627 A164588 * A277672 A015454 A121246

Adjacent sequences:  A022998 A022999 A023000 * A023002 A023003 A023004

KEYWORD

easy,nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified December 5 10:45 EST 2019. Contains 329751 sequences. (Running on oeis4.)