cp's OEIS Frontend

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

%I A023001 #146 Mar 14 2025 15:05:31
%S A023001 0,1,9,73,585,4681,37449,299593,2396745,19173961,153391689,1227133513,
%T A023001 9817068105,78536544841,628292358729,5026338869833,40210710958665,
%U A023001 321685687669321,2573485501354569,20587884010836553,164703072086692425
%N A023001 a(n) = (8^n - 1)/7.
%C A023001 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
%C A023001 {1, 9, 73, 585, 4681, ...} is the binomial transform of A003950. - _Philippe Deléham_, Jul 22 2005
%C A023001 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
%C A023001 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
%C A023001 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
%C A023001 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
%C A023001 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
%C A023001 From _Bernard Schott_, May 01 2017: (Start)
%C A023001 Except for 0, 1 and 73, all the terms are composite because a(n) = ((2^n - 1) * (4^n + 2^n + 1))/7.
%C A023001 For n >= 3, all terms are Brazilian repunits numbers in base 8, and so belong to A125134.
%C A023001 a(3) = 73 is the only Brazilian prime in base 8, and so it belongs to A085104 and A285017. (End)
%H A023001 Vincenzo Librandi, <a href="/A023001/b023001.txt">Table of n, a(n) for n = 0..1000</a>
%H A023001 A. Abdurrahman, <a href="https://arxiv.org/abs/1909.10889">CM Method and Expansion of Numbers</a>, arXiv:1909.10889 [math.NT], 2019.
%H A023001 Carlos M. da Fonseca and Anthony G. Shannon, <a href="https://doi.org/10.7546/nntdm.2024.30.3.491-498">A formal operator involving Fermatian numbers</a>, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
%H A023001 Roger B. Eggleton, <a href="http://dx.doi.org/10.1155/2015/216475">Maximal Midpoint-Free Subsets of Integers</a>, International Journal of Combinatorics Volume 2015, Article ID 216475, 14 pages.
%H A023001 Wolfdieter Lang, <a href="/A023001/a023001.pdf">Notes on certain inhomogeneous three term recurrences.</a> - _Wolfdieter Lang_, Oct 18 2010
%H A023001 Kival Ngaokrajang, <a href="/A023001/a023001_1.pdf">Illustration of initial terms</a>
%H A023001 Quynh Nguyen, Jean Pedersen, and Hien T. Vu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Pedersen/pedersen2.html">New Integer Sequences Arising From 3-Period Folding Numbers</a>, Vol. 19 (2016), Article 16.3.1. See Table 1.
%H A023001 Kalika Prasad, Munesh Kumari, Rabiranjan Mohanta, and Hrishikesh Mahato, <a href="https://arxiv.org/abs/2307.08073">The sequence of higher order Mersenne numbers and associated binomial transforms</a>, arXiv:2307.08073 [math.NT], 2023.
%H A023001 D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, <a href="https://doi.org/10.3390/axioms14030203">On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence</a>, Axioms 14, 203, (2025). See p. 4.
%H A023001 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Repunit.html">Repunit.</a>
%H A023001 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-8).
%F A023001 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
%F A023001 a(n) = A033138(3n-2). - _Alexandre Wajnberg_, May 31 2005
%F A023001 From _Philippe Deléham_, Oct 12 2006: (Start)
%F A023001 a(0) = 0, a(n) = 8*a(n-1) + 1 for n>0.
%F A023001 G.f.: x/((1-8x)*(1-x)). (End)
%F A023001 From _Wolfdieter Lang_, Oct 18 2010: (Start)
%F A023001 a(n) = 7*a(n-1) + 8*a(n-2) + 2, a(0)=0, a(1)=1.
%F A023001 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)
%F A023001 a(n) = Sum_{k=0..n-1} 8^k. - _Doug Bell_, May 26 2017
%F A023001 E.g.f.: exp(x)*(exp(7*x) - 1)/7. - _Stefano Spezia_, Mar 11 2023
%e A023001 From _Zerinvary Lajos_, Jan 14 2007: (Start)
%e A023001 Octal.............decimal
%e A023001 0....................0
%e A023001 1....................1
%e A023001 11...................9
%e A023001 111.................73
%e A023001 1111...............585
%e A023001 11111.............4681
%e A023001 111111...........37449
%e A023001 1111111.........299593
%e A023001 11111111.......2396745
%e A023001 111111111.....19173961
%e A023001 1111111111...153391689
%e A023001 etc. ...............etc. (End)
%e A023001 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
%p A023001 a:=n->sum(8^(n-j),j=1..n): seq(a(n), n=0..20); # _Zerinvary Lajos_, Jan 04 2007
%t A023001 Table[(8^n-1)/7, {n, 0, m}]
%t A023001 LinearRecurrence[{9,-8},{0,1},30] (* _Harvey P. Dale_, Feb 12 2015 *)
%o A023001 (Sage) [lucas_number1(n,9,8) for n in range(0, 21)] # _Zerinvary Lajos_, Apr 23 2009
%o A023001 (Sage) [gaussian_binomial(n,1,8) for n in range(0,21)] # _Zerinvary Lajos_, May 28 2009
%o A023001 (Magma) [(8^n-1)/7: n in [0..20]]; // _Vincenzo Librandi_, Sep 17 2011
%o A023001 (Maxima) A023001(n):=floor((8^n-1)/7)$
%o A023001 makelist(A023001(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o A023001 (PARI) a(n)=(8^n-1)/7 \\ _Charles R Greathouse IV_, Mar 22 2016
%o A023001 (GAP)
%o A023001 A023001:=List([0..10^2],n->(8^n-1)/7); # _Muniru A Asiru_, Oct 03 2017
%Y A023001 Cf. A007556, A003950, A001158, A033138.
%Y A023001 Cf. A125134, A085104, A285017, A220571, A053696.
%K A023001 easy,nonn
%O A023001 0,3
%A A023001 _David W. Wilson_