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A002452
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(9^n - 1)/8.
(Formerly M4733 N2025)
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69
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0, 1, 10, 91, 820, 7381, 66430, 597871, 5380840, 48427561, 435848050, 3922632451, 35303692060, 317733228541, 2859599056870, 25736391511831, 231627523606480, 2084647712458321, 18761829412124890, 168856464709124011
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OFFSET
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0,3
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COMMENTS
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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 DELEHAM, Jul 22 2005
Numbers in base 9: 1, 11, 111, 1111, 11111, 111111,1111111, etc. [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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). [From 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). [From Milan Janjic, Feb 21 2010]
Contribution 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)
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REFERENCES
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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.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..300
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Repunit
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
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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]
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MAPLE
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a:=n->sum(9^(n-j), j=1..n): seq(a(n), n=1..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 04 2007
A002452:=1/(9*z-1)/(z-1); [Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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lst={}; Do[p=(9^n-1)/8; AppendTo[lst, p], {n, 0, 5!}]; lst [From Vladimir Joseph Stephan Orlovsky, Sep 29 2008]
(9^# & /@ Range[0, 18] // Accumulate) (* Added by Ant King Jan 06 2011)
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PROG
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(Sage) [lucas_number1(n, 10, 9) for n in xrange(1, 20)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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 */
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CROSSREFS
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Right-hand column 1 in triangle A008958.
Cf. A217094, A011540, A052382.
Sequence in context: A143572 A002739 A079928 * A096261 A015455 A110410
Adjacent sequences: A002449 A002450 A002451 * A002453 A002454 A002455
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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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
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STATUS
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approved
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