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
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
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
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) = 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
KEYWORD
nonn,easy
AUTHOR
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