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A023001 (8^n - 1)/7. 64
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 (Mod[A007556[A0023001(n)],2]=1). - Farideh Firoozbakht, Jun 13 2003

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

{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). [From 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). [From 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. [From Wolfdieter Lang, Oct 18 2010]

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

LINKS

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

Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [From Wolfdieter Lang, Oct 18 2010]

Eric Weisstein's World of Mathematics, Repunit.

Index entries for sequences related to 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(0)=0, a(n)=8*a(n-1)+1 for n>0 . G.f.:x/((1-8x)*(1-x)) - Philippe Deléham, Oct 12 2006

Contribution 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 G. Detlefs. See the W. Lang comment and link. (End)

EXAMPLE

Octal.............decimal (comment from Zerinvary Lajos, Jan 14 2007):

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

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

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

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

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

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

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

1111111.........299593

11111111.......2396745

111111111.....19173961

1111111111...153391689

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

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}]

PROG

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

(Sage) [gaussian_binomial(n, 1, 8) for n in xrange(0, 21)] # [From 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 */

CROSSREFS

Cf. A007556.

Sequence in context: A126641 A081627 A164588 * A015454 A121246 A086226

Adjacent sequences:  A022998 A022999 A023000 * A023002 A023003 A023004

KEYWORD

easy,nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified November 23 18:50 EST 2014. Contains 249865 sequences.