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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Examples

			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
		

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
• Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
• 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.
• D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 4.
• 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).

Crossrefs

Right-hand column 1 in triangle A008958.

Cf. A217094, A011540, A052382, A125857.

Cf. A125134, A000217.

Programs

Magma

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

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 *)

Maxima

A002452(n):=floor((9^n-1)/8)$
makelist(A002452(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI

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

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

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