A002452 a(n) = (9^n - 1)/8. (Formerly M4733 N2025)

0, 1, 10, 91, 820, 7381, 66430, 597871, 5380840, 48427561, 435848050, 3922632451, 35303692060, 317733228541, 2859599056870, 25736391511831, 231627523606480, 2084647712458321, 18761829412124890, 168856464709124011, 1519708182382116100, 13677373641439044901, 123096362772951404110

Offset

0,3

Comments

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)

For n > 0, 4*a(n) is the total number of holes in a certain triangle fractal (start with 9 triangles, 4 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Feb 21 2015

For n > 0, a(n) is the sum of the numerators and denominators of the reduced fractions 0 < (b/3^(n-1)) < 1 plus 1. Example for n=3 gives fractions 1/9, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, and 8/9 plus 1 has sum of numerators and denominators +1 = a(3) = 91. - J. M. Bergot, Jul 11 2015

Except for 0 and 1, all terms are Brazilian repunits numbers in base 9, so belong to A125134. All these terms are composite because a(n) is the ((3^n - 1)/2)-th triangular number. - Bernard Schott, Apr 23 2017

These are also the second steps after the junctions of the Collatz trajectories of 2^(2k-1)-1 and 2^2k-1. - David Rabahy, Nov 01 2017

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.

Links

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

Kival Ngaokrajang, Illustration of initial terms

Vladimir Pletser, Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers, arXiv:1409.7969 [math.NT], 2014.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

A. G. Shannon, Letter to N. J. A. Sloane, Dec 06 1974

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

Eric Weisstein's World of Mathematics, Repunit

Index entries for linear recurrences with constant coefficients, signature (10,-9).

Formula

From Philippe Deléham, Mar 13 2004: (Start)

a(n) = 9*a(n-1) + 1; a(1) = 1.

G.f.: x / ((1-x)*(1-9*x)). (End)

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

Sum_{n>0} a(n)*(-1)^(n+1)*x^(2*n+1)/(2*n+1)! = (1/6)*sin(x)^3. - 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) = Sum_{k=0..n-1} 9^k. - Doug Bell, May 26 2017

E.g.f.: exp(5*x)*sinh(4*x)/4. - Stefano Spezia, Mar 11 2023

Example

a(4) = (9^4 - 1)/8 = 820 = 1111_9 = (1/2) * 40 * 41 is the ((3^4 - 1)/2)-th = 40th triangular number. - Bernard Schott, Apr 23 2017

Maple

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

Mathematica

(9^# & /@ Range[0, 18] // Accumulate) (* Ant King, Jan 06 2011 *)
LinearRecurrence[{10, -9}, {0, 1}, 30] (* Harvey P. Dale, Sep 23 2018 *)

Prog

(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.

Cf. A125134, A000217.

Sequence in context: A079928 A346230 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