A023001 a(n) = (8^n - 1)/7.

0, 1, 9, 73, 585, 4681, 37449, 299593, 2396745, 19173961, 153391689, 1227133513, 9817068105, 78536544841, 628292358729, 5026338869833, 40210710958665, 321685687669321, 2573485501354569, 20587884010836553, 164703072086692425

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.

Kalika Prasad, Munesh Kumari, Rabiranjan Mohanta, and Hrishikesh Mahato, The sequence of higher order Mersenne numbers and associated binomial transforms, arXiv:2307.08073 [math.NT], 2023.

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

E.g.f.: exp(x)*(exp(7*x) - 1)/7. - Stefano Spezia, Mar 11 2023

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 range(0, 21)] # Zerinvary Lajos, Apr 23 2009
(Sage) [gaussian_binomial(n, 1, 8) for n in range(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: A346229 A081627 A164588 * A277672 A015454 A343353

Adjacent sequences: A022998 A022999 A023000 * A023002 A023003 A023004

Keyword

easy,nonn

Author

David W. Wilson

Status

approved