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

Author

David W. Wilson

Keywords

• easy
• nonn

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)

Examples

			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
		

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.
• Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
• 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.
• D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 4.
• Eric Weisstein's World of Mathematics, Repunit.
• Index entries for linear recurrences with constant coefficients, signature (9,-8).

Crossrefs

Cf. A007556, A003950, A001158, A033138.

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

Programs

GAP

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

Magma

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

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 *)

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

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

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