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A002452 (9^n - 1)/8.
(Formerly M4733 N2025)
74
0, 1, 10, 91, 820, 7381, 66430, 597871, 5380840, 48427561, 435848050, 3922632451, 35303692060, 317733228541, 2859599056870, 25736391511831, 231627523606480, 2084647712458321, 18761829412124890, 168856464709124011, 1519708182382116100, 13677373641439044901, 123096362772951404110 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Comment from David W. Wilson: Numbers triangular, differences square.

To be precise, the differences are the squares of the powers of three with positive indices. Hence a(n+1)-a(n) = (A000244(n+1))^2 = A001019(n+1). (Added by Ant King Jan 05 2011)

Partial sums of A001019. This is m-th triangular number, where m is partial sums of A000244. a(n) = A000217(A003462(n)). - Lekraj Beedassy, May 25 2004

With offset 0, binomial transform of A003951. - Philippe Deléham, Jul 22 2005

Numbers in base 9: 1, 11, 111, 1111, 11111, 111111,1111111, etc. - Zerinvary Lajos, Apr 26 2009

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)=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]:=10, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=2, a(n-1)=(-1)^n*charpoly(A,1). - Milan Janjic, Feb 21 2010

From Hieronymus Fischer, Jan 30 2013: (Start)

Least index k such that A052382(k) >= 10^(n-1), for n>0.

Also index k such that A052382(k) = (10^n-1)/9, n>0.

A052382(a(n)) is the least zerofree number with n digits, for n>0.

For n>1: A052382(a(n)-1) is the greatest zerofree number with n-1 digits. (End)

REFERENCES

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 36.

M. Ward, Note on divisibility sequences, Bull. Amer. Math. Soc., 42 (1936), 843-845.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Repunit

Index entries for sequences related to linear recurrences with constant coefficients

FORMULA

a(n) = 9*a(n-1) + 1; a(1) = 1. G.f.: x / ((1-x)*(1-9*x)). - Philippe Deléham, Mar 13 2004

a(n) = 10*a(n-1) - 9*a(n-2). - Ant King, Jan 05 2011

a(n) = floor(A000217(3^n)/4) - A033113(n-1). - Arkadiusz Wesolowski, Feb 14 2012

E.g.f.: 1/6*sin(x)^3 = sum{n>0, a(n)*(-1)^(n+1)*x^(2*n+1)/(2*n+1)!}. - Vladimir Kruchinin, Sep 30 2012

a(n) = A011540(A217094(n-1)) - A217094(n-1) + 2, n>0. - Hieronymus Fischer, Jan 30 2013

a(n) = 10^(n-1) + 2 - A217094(n-1). - Hieronymus Fischer, Jan 30 2013

a(n) = det(|v(i+2,j+1)|, 1 <= i,j <= n-1), where v(n,k) are central factorial numbers of the first kind with odd indices (A008956) and n>0. -Mircea Merca, Apr 06 2013

a(n) = A125857(n)/2. - Vladimir Pletser, Mar 29 2014

MAPLE

a := n -> add(9^(n-j), j=1..n): seq(a(n), n=0..19); # Zerinvary Lajos, Jan 04 2007

A002452 := 1/(9*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation.

MATHEMATICA

lst={}; Do[p=(9^n-1)/8; AppendTo[lst, p], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 29 2008 *)

(9^# & /@ Range[0, 18] // Accumulate) (* Ant King, Jan 06 2011 *)

PROG

(Sage) [lucas_number1(n, 10, 9) for n in xrange(1, 20)] # Zerinvary Lajos, Apr 26 2009

(MAGMA) [(9^n - 1)/8 : n in [0..25]]; // Vincenzo Librandi, Jun 01 2011

(PARI) a(n)=9^n>>3 \\ Charles R Greathouse IV, Jul 25 2011

(Maxima) A002452(n):=floor((9^n-1)/8)$

makelist(A002452(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */

CROSSREFS

Right-hand column 1 in triangle A008958.

Cf. A217094, A011540, A052382, A125857.

Sequence in context: A002739 A079928 A231412 * A096261 A015455 A110410

Adjacent sequences:  A002449 A002450 A002451 * A002453 A002454 A002455

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004

Offset changed from 1 to 0 and added 0  by Vincenzo Librandi, Jun 01 2011

STATUS

approved

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Last modified July 24 15:58 EDT 2014. Contains 244896 sequences.